Suggestions for how to prepare to teach this unit, Internalization of Standards via the Unit Assessment, The central mathematical concepts that students will come to understand in this unit, Terms and notation that students learn or use in the unit, The materials, representations, and tools teachers and students will need for this unit, Topic A: Right Triangle Properties and Side-Length Relationships. Consider a 30-60-90 triangle with the longer leg measuring 9 inches. Vertical side b is 1 unit. Lesson 11 Practice Problems The right triangles are drawn in the coordinate plane, and the coordinates of their vertices are labeled. You need to see someone explaining the material to you. Help! In this warm-up, students compare four triangles. The triangle on the left has the square labels a squared equals 16 aligned to the bottom horizontal leg and b squared equals 10 aligned to the left leg. 11. You may not send out downloaded content to any third party, including BOCES districts, to copy and or bind downloaded content. Lesson 1 3. Please do not copy or share the Answer Keys or other membership content. The square labeled c squared equals 17 is attached to the hypotenuse. Restart your browser. Direct link to AHsciencegirl's post Good point, let's estimat, Posted 3 years ago. The height of the triangle is 1. The pilot spots a person with an angle of depression . In order to continue to provide high quality mathematics resources to you and your students we respectfully request that you do not post this or any of our files on any website. 45-45-90 triangles are right triangles whose acute angles are both. On this page you will find some material about Lesson 26. Rationalize the denominator. - %PDF-1.5 % The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). One example is: sin of 1 angle (in the right triangle) = opposite over hypotenuse. from Lesson 7-4 that apply only to right triangles. The swing ropes are. Collaborate slope triangles are related. Direct link to Siena's post Can't you just use SOH CA, Posted 3 years ago. Use side and angle relationships in right and non-right triangles to solve application problems. %%EOF F.TF.A.4 Maybe the answer wouldn't differ that much but it might make it a little more challenging to figure out. With 45-45-90 and 30-60-90 triangles you can figure out all the sides of the triangle by using only one side. If you start with x3 = 18, divide both sides by 3 to get x = 18/3, but since we do not like roots in the denominator, we then multiply by 3/3 to get 183/(3*3) = 18 3/3=63. . Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. You can make in-house photocopies of downloaded material to distribute to your class. Home > INT2 > Chapter 6 > Lesson 6.1.1 > Problem 6-6. I know that to get the answer I need to multiply this by the square root of 3 over 2. The square of the hypotenuse is equal to the sum of the squares of the legs. F.TF.B.7 Shouldn't we take in account the height at which the MIB shoots its laser. This is a "special" case where you can just use multiples: 3 - 4 - 5 G.SRT.B.4 3 This site includes public domain images or openly licensed images that are copyrighted by their respective owners. This is not correct. The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group. Side B C is unknown. How do we use our calculator to find an unknown angle in a right triangle if two sides are given? f;XqvFOh| -<5, l"G3bsK}^";@-.;{+\c]sg{VNj~@ZDof HWtt4Tt4pE .i 432libPq0M2aT!rJwTr}x$000``c z \Oi(Yxb@ t Verify experimentally the properties of rotations, reflections, and translations: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. The special properties of both of these special right triangles are a result of the. Direct link to 91097027's post do i have to be specific, Posted 4 years ago. Verify experimentally the properties of rotations, reflections, and translations: 8.G.A.4 Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Direct link to NightmareChild's post I agree with Spandan. Prove the Pythagorean identity sin() + cos() = 1 and use it to find sin(), cos(), or tan() given sin(), cos(), or tan() and the quadrant of the angle. The lengths of the sides of a triangle are 3x, 5x - 12 and x + 20 Find the value of x so that the triangle is isosceles. What is the relationship between an angle of depression and an angle of elevation? What is the value of sine, cosine, and tangent? Creative Commons Attribution 4.0 International License (CC BY 4.0), https://openupresources.org/math-curriculum/. Side c slants downward and to the right. After 12 minutes of quiet think time, ask partners to discuss their strategies and then calculate the values. The ratios come straight from the Pythagorean theorem. Look at the formula of each one of them. Winter 2023, GEOMETRY 123A what can i do to not get confused with what im doing ? One of the main goals in this unit is a deep understanding of the unit circle. Each of the vertices of the inside square divides the side lengths of the large square into two lengths: 8 units and 6 units creating 4 right triangles.
. Direct link to anthony.lozano's post what can i do to not get , Posted 6 years ago. The Sine, Cosine, and Tangent are three different functions. WHY. Use similarity criteria to generalize the definition of sine to all angles of the same measure. Unit 4: Right Triangles and Trigonometry. For sine and cosine, yes because the hypotenuse will always be the longest side, but for tangent, it does not have to be, either the opposite or the adjacent could be longer than the other. Ask each group to share one reason why a particular triangledoes not belong. For example, see x4 y4 as (x) (y), thus recognizing it as a difference of squares that can be factored as (x y)(x + y). - The following assessments accompany Unit 4. a link to a video lesson. The trigonometric ratios sine, cosine, and tangent can have different signs, negative or positive, depending in which quadrant of the coordinate plane the angle and right triangle lie. how do i know to use sine cosine or tangent? hbbd```b``"@$z^ F.TF.B.6 In China, a name for the same relationship is the Shang Gao Theorem. Vertical side b is 3 units. If we add the areas of the two small squares, we get the area of the larger square. [How can we find these ratios using the Pythagorean theorem? hb```l eae2SIU A square is drawn using each side of the triangles. A.SSE.A.2 a. Write W, X, Y, or Z. Identify these in two-dimensional figures. The height of the triangle is 2. I agree with Spandan. Direct link to mud's post wow, thanks :), Posted 4 years ago. Recognize and represent proportional relationships between quantities. Diagonal side c slants downward and to the right and the triangle has a height of 1 unit. Side b and side c are equal in length. Mathematics Textbook Correlation to the 2016 Grade Eight Mathematics Standards of Learning and Curriculum Framework Grade Eight Mathematics 12 of 29 Virginia Department of Education 2017 Page: M4-75A Lesson: 3. The swing will be closer than 2.75 meters at the bottom of the arc. Look for and express regularity in repeated reasoning. Congruent figures. Chapter 8 - Right Triangle Trigonometry Answer Key CK-12 Geometry Concepts 2 8.2 Applications of the Pythagorean Theorem Answers 1. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. $B9K=>"-b)FC!&4 NS-xIC(XV%gOcB"hc%C,x/_ 1?gz>f8,,iIO6g/bT+d|.z5gg9"H9yP1FlRINgb:&R5!'O}`$_UBDXG16k_ ${ x2ZlTh[hwwc>R;`O" t9}!H}1LEsUS6!H4Y;O,8|(Wwy X20 / Problem 1.1 BC= B C = Round your answer to the nearest hundredth. You would even be able to calculate the height the agent is holding his gun at with stretched arms when you know the angle he's keeping his arms at, his arm length and the length from his shoulders to the ground. Pretend that the short leg is 4 and we will represent that as "x." And we are trying to find the length of the hypotenuse side and the long side. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. Use the graph to discover how. No 4. Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Solve applications involving angles of rotation. Attend to precision. UNIT 5 TEST: Trigonometric Functions PART 2 . Prove theorems about triangles. For our full Disclaimer of Warranties, please see our Single User License Agreement Here. - Angle A B C is forty degrees. Side A B is six units. N.RN.A.2 Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. Side B C is six units. U08.AO.02 - Right Triangle Trigonometry Practice RESOURCE ANSWER KEY EDITABLE RESOURCE EDITABLE KEY Get Access to Additional eMath Resources Register and become a verified teacher for greater access. The length of both legs are k units. Thats why we may do the following (and we ask that you agree): SATISFACTION GUARANTEED. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. If you want to get the best homework answers, you need to ask the right questions. Chapter 6 congruent triangles answer key - II. One of the best known mathematical formulas is Pythagorean Theorem, which provides us with the relationship between the sides in a right triangle. In a triangle 30-60-90, if I am given the long side as an integer, how can I derive the calculation of the other sides? Diagonal side c slants downward and to the right and the triangle has a height of 3 units. Pause, rewind, replay, stop follow your pace! 2016-2017 Congruency, Similarity, Right Triangles, and Trigonometry - Answer Key 3 MAFS.912.G-CO.1.1 EOC Practice Level 2 Level 3 Level 4 Level 5 uses definitions to choose examples and non-examples uses precise definitions that are based on the undefined notions of point, line, distance along a line, and distance around a circular arc PLEASE, NO SHARING. TECHNICAL SUPPORT: If you are having trouble logging in or accessing your materials, or if your downloaded materials wont open or are illegible, please notify us immediately by email at[emailprotected]so we can get it fixed. This directly reflects work students have done previously for finding the length of a diagonal on a grid. Unit 6 triangles and congruence lesson 1 answer key - Unit 6-Triangles & Congruence. Compare two different proportional relationships represented in different ways. Then complete the sentences. We encourage you to try the Try Questions on your own. We believe in the quality and value of our products and services, and we work hard to make sure they work well and are free of bugs. A right triangle A B C. Angle A C B is a right angle. Yes, but special right triangles have constant ratios, so if you learn how to do this, you can get answers faster. Solve for missing sides of a right triangle given the length of one side and measure of one angle. The design of the chair swing ride. To find a triangle's area, use the formula area = 1/2 * base * height. Chapter 1 - Introduction to Trigonometry Answer Key CK-12 Trigonometry Concepts 3 1.3 Pythagorean Theorem to Classify Triangles Answers 1. - order now. The triangle on the right has the square labels a squared equals 9 and b squared equals 9 attached to each of the legs. 1 2 3 831 Use a separate piece of . Then tell students that the Pythagorean Theorem says: If \(a\), \(b\), and \(c\) are the sides of a right triangle, where \(c\) is the hypotenuse, then. 8.G.B.7 Find the angle measure given two sides using inverse trigonometric functions. Direct link to Esa Abuzar's post if I get 30.1 degrees, is, Posted 3 years ago. Unit 8 right triangles and trigonometry test answer key. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. Students gain practice with determining an appropriate strategy for solving right triangles. Here is a diagram of an acute triangle . Some squares are intentionally positioned so that students won't be able to draw squares and must find other ways to find the side lengths. endstream endobj 1779 0 obj <>/Metadata 152 0 R/Pages 1776 0 R/StructTreeRoot 184 0 R/Type/Catalog>> endobj 1780 0 obj <>/MediaBox[0 0 612 792]/Parent 1776 0 R/Resources<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI]/XObject<>>>/Rotate 0/StructParents 0/Tabs/S/Type/Page>> endobj 1781 0 obj <>stream Use the tangent ratio of the angle of elevation or depression to solve real-world problems. Unit 5 Right Triangles TEST REVIEW Solutions. Create a free account to access thousands of lesson plans. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Side b and side c are equal in length. If you create a modified assignment using a purchased editable file, please credit us as follows on all assignment and answer key pages: Use your feedback to make improvements to our products and services and even launch new products and services, with the understanding that you will not be paid or own any part of the new or improved products and services (unless we otherwise agree in writing ahead of time). Students develop the algebraic tools to perform operations with radicals. If you are a school, please purchase a license for each teacher/user. Register and become a verified teacher for greater access. Solve applications involving angles of elevation and depression. That is an interesting point that I hadn't considered, but not what the question is asking. It is important to note that this relationship does not hold for all triangles. Side b slants upward and to the left. He explains that, two straight lengths of wire are placed on the ground, forming vertical angles. Please click the link below to submit your verification request. In future lessons, you will learn some ways to explain why the Pythagorean Theorem is true for any right triangle. WeBWorK. F.TF.C.8 See the image attribution section for more information. junio 12, 2022. abc news anchors female philadelphia . Math Questions Solve Now Chapter 6 congruent triangles answer key . Lesson 6 Homework Practice. This will rely heavily on the use of special right triangles. We will use this opportunity to make connections with other concepts. We value your feedback about our products and services. For Example-. Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and use the unit circle to express the values of sine, cosine, and tangent for -x, +x, and 2-x in terms of their values for x, where x is any real number. The triangle on the left has the square labels a squared equals 16 and b squared equals 9 attached to each of the legs. Find a. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Answer keys are for teacher use only and may not be distributed to students. IM 68 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. We believe in the value we bring to teachers and schools, and we want to keep doing it. The trig functions give outputs in terms of the ratios of two sides of a triangle when we feed them the input of an angle measure. Sign in Students develop an understanding of right triangles through an introduction to trigonometry, building an appreciation for the similarity of triangles as the basis for developing the Pythagorean theorem. The triangle must be a right triangle with an altitude to the hypotenuse. 1 . Adaptations and updates to IM 68 Math are copyright 2019by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). A 200 meter long road travels directly up a 120 meter tall hill. Look for and express regularity in repeated reasoning. Direct link to april_oh_'s post I use this trick on 30, 6, Posted a year ago. Triangle D, right, legs = 3,4. hypotenuse = 5. Explore our childs talent throught the wonderful experience of painting. Then calculate the area and perimeter of the triangle. Prove the Pythagorean identity sin() + cos() = 1 and use it to find sin(), cos(), or tan() given sin(), cos(), or tan() and the quadrant of the angle. when working out the inverse trig, is the bigger number always on the bottom? Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. A right triangle consists of two legs and a hypotenuse. Section 2.3: Applications of Static Trigonometry. Direct link to Aryan's post What is the difference be, Posted 6 years ago. Course Hero is not sponsored or endorsed by any college or university. Students use similarity to prove the Pythagorean theorem and the converse of the Pythagorean theorem. The, Posted 6 years ago. G.SRT.B.5 This includes school websites and teacher pages on school websites. 8.G.A.1 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. What is the sum of the angles of a triangle? Special Triangle: This is a triangle whose angles are , and . How does the length of the hypotenuse in a right triangle compare to the lengths of the legs? 2. what is the value of x and y? 45 5. Direct link to Trevor Amrhannah Davis's post My problem is that I do n, Posted 3 years ago. if I get 30.1 degrees, is it still a special triangle. No, but it is approximately a special triangle. - When you use this site, you are agreeing to comply with these Terms & Conditions and our Single User License Agreement. Ask students to check that the Pythagorean Theorem is true for these triangles. CCSS.MATH.PRACTICE.MP2 289.97 u2 3. Side A B is eight units. The hypotenuse is opposite the right angle. Are special right triangles still classified as right triangles? Harsh. Direct link to Markarino /TEE/DGPE-PI1 #Evaluate's post Boy, I hope you're still , Posted 5 years ago. Some segments are congruent to others whose lengths are already known. G.SRT.B.4 Theorems about right triangles (e.g., Pythagorean theorem, special right triangles, and use of an altitude to make right triangles) give additional tools for finding missing measures. sharwood's butter chicken slow cooker larry murphy bally sports detroit lesson 1: the right triangle connection answer key. If students do not see these patterns, dont give it away. The Pythagorean Theorem: Ex. Direct link to Thien D Ho's post Look at the formula of ea, Posted 2 years ago. Using these materials implies you agree to our terms and conditions and single user license agreement. Unit 8 Right Triangles And Trigonometry Homework 1 Answers Key*If c^2 = a^2 + Bell: Homework 1: Pythagorean Theorem and its Converse - This is a 2-page . (from Coburn and Herdlicks Trigonometry book), Section 2.2: Solving Right Triangles, and. Solve applications involving angles of rotation. - Many times the mini-lesson will not be enough for you to start working on the problems. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. How does the length of the hypotenuse in a right triangle compare to the lengths of the legs? Please do not post the Answer Keys or other membership content on a website for others to view. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Theanglemade bythelineof sight ofanobserveronthegroundtoapointabovethe horizontaliscalled the angle of elevation. A 45 45 90 triangle is isosceles. Pacing: 21instructional days (19 lessons, 1 flex day, 1 assessment day). The hypotenuse of a 45-45-90 triangle measures cm. For more information, check the. One key thing for them to notice is whether the triangleis a right triangle or not. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Topic C: Applications of Right Triangle Trigonometry. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. How is this related to finding the positive solution to the equation, Visit a tutor. Get math help online by chatting with a tutor or watching a video lesson. By using the Pythagorean Theorem, we obtain that. What are the sides of a right triangle called? Explain a proof of the Pythagorean Theorem and its converse. The OpenLab is an open-source, digital platform designed to support teaching and learning at City Tech (New York City College of Technology), and to promote student and faculty engagement in the intellectual and social life of the college community. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Prove the Laws of Sines and Cosines and use them to solve problems. CCSS.MATH.PRACTICE.MP6 Ask students to indicate when they have noticed one triangle that does not belong and can explain why. Howard is designing a chair swing ride. Define and calculate the cosine of angles in right triangles. No Is this a right triangle: a=4, b=6, c=9 yes Is this a right triangle: a=5 b=12 c=13 a triangle where one angle is guaranteed to be 90 degrees. This will help you with your trig skills. 's':'']}, GEOMETRY UNIT 5 Notice that for these examples of right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. You are correct that it is an arc. Solve a right triangle given one angle and one side. Posted 6 years ago. Ask selected students to share their reasoning. Model with mathematics. - If you know the 30-degree side of a 30-60-90 triangle the 60-degree side is root 3 times larger and the hypotenuse is twice as long. Side c slants downward and to the right. It can be also used as a review of the lesson. 8.G.B.6 8 spiritual secrets for multiplying your money. 10. The purpose of this task is for students to thinkabout the relationships between the squares of theside lengths of triangles as a leadup to the Pythagorean Theorem at the end of this lesson. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Either the problem will tell you which angle is the reference angle or it will give two sides and you can choose which of the two acute angles you can use as the reference angle. 1778 0 obj <> endobj Round your answers to the nearest tenth. G.CO.A.1 G.SRT.B.4 2 Define and prove the Pythagorean theorem. The hypotenuse of a right triangle is the longest side. Pythagorean Theorem: In a right triangle, if the legs measure and and the hypotenuse measures , then. Would the answer to this problem be 36 (square root of 3 times the square root of 3 to get 3, 2 times 6 to get 12, and 12 times 3 to get 36)? Take your time to do them, and check your answer by clicking on the Show Answer tab. Side b slants upward and to the left. Connexus Connections Academy (Connections Academy Online, MCA)'s GEOMETRY department has 8 courses in Course Hero with 92 documents and 62 answered questions. Doing the homework is an essential part of learning. Tell students they will use their strategies to determine the side lengths of several triangles in the activity. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. The two legs are equal. 's':'']}, {[ course.numQa ]} Q&A{[course.numQa>1? The square labeled c squared equals 18 is aligned with the hypotenuse. Similar Right Triangles To Find Slope Teaching Resources . A brief review will help you boost your confidence to start the new lesson, and thats perfectly fine. Unit 4 Homework 4 Congruent Triangles Answer Key Athens. Description:Three right triangles are indicated. To make this example correct the 2,75 meters needs to be applied to the point where the swing is parallel to the supporting pole. Define and calculate the sine of angles in right triangles. This skill is extended in Topic D, the Unit Circle, where students are introduced to the unit circle and reference angles.
. Lesson Map Topic A: Right Triangle Properties and Side-Length Relationships 1 Define the parts of a right triangle and describe the properties of an altitude of a right triangle. I am so confusedI try my best but I still don't get it . 4. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Detailed Answer Key. It is time to do the homework on WeBWork: When you are done, come back to this page for the Exit Questions. Special Right Triangles Worksheet Answer Key.pdf - Google Drive . "YnxIzZ03]&E$H/cEd_ O$A"@U@ This triangle is special, because the sides are in a special proportion. CCSS.MATH.PRACTICE.MP3 REMEMBER One Pythagorean identity states that sin 2 + cos = 1. Posted 6 years ago. You can view more similar questions or ask a . To read the Single User License Agreement, please clickHERE. . So in addition to agreeing not to copy or share, we ask you: This assignment is a teacher-modified version of [eMATHTitle] Copyright 201xeMATHinstruction, LLC, used by permission. im so used to doing a2+b2=c 2 what has changed I do not understand. F.TF.B.5 Given sin = _1 in Quadrant IV, determine 3 cos . and and and Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. 30-60-90 triangles are right triangles whose acute angles are. Lesson 6. Purpose of each question: spiral, foundational, mastery, developing, Strategies and representations used in daily lessons, Relationship to Essential Understandings of unit, Notice the progression of concepts through the unit using Unit at a Glance.. A right triangle is a triangle with a right angle. Direct link to jinseo.park's post Are special right triangl, Posted 4 years ago. . They do not have a value outright, it would be like trying to ask what the value of f(x) = x + 1 is. Use the structure of an expression to identify ways to rewrite it. there is a second square inside the square. These Terms & Conditions present some of the highlights of the Single User License Agreement in plain English, but its a good idea to look at the complete Single User License Agreement, too, because by checking the box below and proceeding with your purchase you are agreeing to both these Terms & Conditions and the Single User License Agreement. An isosceles triangle is. Thank you for using eMATHinstruction materials. Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Use appropriate tools strategically. In this activity, studentscalculate the side lengthsof the triangles by both drawing in tilted squares and reasoning about segments that must be congruent to segments whose lengths are known. Complete each statement with always, sometimes or never. 1. Teachers with a valid work email address canclick here to register or sign in for free access to Cool-Downs. Direct link to sydney's post How can you tell if a tri, Posted 4 years ago. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. This is true, but, if no student points it out, note that \(3 = \sqrt{9}\), and so the strategy of drawing in a square still works. If you already have a plan, please login. However, the key to the question is the phrase "in full swing". So, if you know sin of that angle, and you also know the length of the opposite. Triangle C, right, legs = 1,8. hypotenuse = square root 65. Let's find, for example, the measure of. 24/7 help. Angle B A C is unknown. You will also find one last problem. Verify algebraically and find missing measures using the Law of Cosines. b. d. Use a straightedge to draw squares on each side of the triangle. Your friend claims that two isosceles triangles triangle ABC and triangle DEF are congruent if two corresponding sides are congruent. if you know the 60-degree side of a 30-60-90 triangle the 30-degree side is root 3 times smaller and the hypotenuse is 2/root 3 times longer.