Using Corresponding Parts of Congruent Triangles
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1 4-4 Using orresponding arts of ongruent riangles ontent tandards G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. lso G..12 bjective o use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent Is congruent to GHI? How do you know? How does H help you solve this problem? HI I With,,, and, you know how to use three congruent parts of two triangles to show that the triangles are congruent. nce you know that two triangles are congruent, you can make conclusions about their other corresponding parts because, by definition, corresponding parts of congruent triangles are congruent. ssential Understanding If you know two triangles are congruent, then you know that every pair of their corresponding parts is also congruent. roblem 1 roving arts of riangles ongruent In the diagram, which congruent pair is not marked? he third angles of both triangles are congruent. ut there is no congruence rule. o, find a congruent pair of sides. Given:, rove: Given eflexive roperty of Given heorem orresp. parts of are. Got It? 1. Given:, rove: 244 hapter 4 ongruent riangles
2 Which congruency rule can you use? You have information about two pairs of angles. Guess-andcheck and. roblem 2 roving riangle arts ongruent to easure istance easurement hales, a Greek philosopher, is said to have developed a method to measure the distance to a ship at sea. He made a compass by nailing two sticks together. tanding on top of a tower, he would hold one stick vertical and tilt the other until he could see the ship along the line of the tilted stick. With this compass setting, he would find a landmark on the shore along the line of the tilted stick. How far would the ship be from the base of the tower? Given: and are right angles, rove: tatements easons 1) 1) Given 2) 2) eflexive roperty of ongruence 3) and are right angles. 3) Given 4) 4) ll right angles are congruent. 5) 5) ostulate 6) 6) orresponding parts of are. he distance between the ship and the base of the tower would be the same as the distance between the base of the tower and the landmark. Got It? 2. a. Given:, is the midpoint of rove: b. easoning If the landmark were not at sea level, would the method in roblem 2 work? xplain. esson 4-4 Using orresponding arts of ongruent riangles 245
3 esson heck o you know HW? ame the postulate or theorem that you can use to show the triangles are congruent. hen explain why the statement is true U U o you U? 3. easoning How does the fact that corresponding parts of congruent triangles are congruent relate to the definition of congruent triangles? 4. rror nalysis Find and correct the error(s) in the proof. Given: H H, rove: H is the midpoint of. : H H because it is given. because it is given. H H because vertical angles are congruent. o, H H by ostulate. ince corresponding parts of congruent triangles are congruent, H H. y the definition of midpoint, H is the midpoint of. HI I H ractice and roblem-olving xercises I ractice 5. eveloping ell why the two triangles are congruent. Give the congruence statement. hen ee roblem 1. list all the other corresponding parts of the triangles that are congruent. 6. Given:, rove: 7. Given:, rove: HI ee roblem eveloping balalaika is a stringed instrument. rove that the bases of the balalaikas are congruent. Given: Y, Y, Y rove: Y : It is given that two angles and the included side of one triangle are congruent to two angles and the included side of the other. o, a. Y by b.. Y because c.. Y 246 hapter 4 ongruent riangles
4 pply 9. Given:, rove: 10. Given: Y Y,, rove: Y easoning opy and mark the figure to show the given information. xplain how you would prove Q. 11. Given: Q, bisects Q 12. Given: is the perpendicular bisector of Q. 13. Given: Q, bisects Q 14. hink bout a lan he construction of a line perpendicular to line through point on line is shown. xplain why you can conclude that is perpendicular to. How can you use congruent triangles to justify the construction? Which lengths or distances are equal by construction? Q 15. Given:, bisects rove:, bisects 16. Given:, bisects at, is on rove: 17. onstructions he construction of congruent to given is shown. F because they are congruent radii. F because both arcs have the same compass settings. xplain why you can conclude that. F 18. Given:, F, F, F rove: F 19. Given: Q, Q rove: Q bisects. Q esson 4-4 Using orresponding arts of ongruent riangles 247
5 20. esigns angoli is a colorful design pattern drawn outside houses in India, especially during festivals. Vina plans to use the pattern at the right as the base of her design. In this pattern, U, V, and Q bisect each other at. 6, U 12, U V, U, and Q. What is the perimeter of the hexagon? hallenge In the diagram at the right, and. 21. rove: is the midpoint of. 22. rove: tandardized est rep / For xercises 23 and 24, use the diagram at the right. and bisects and. 23. uppose 17 and 5. What is the length of? 24. uppose m 6 4, and m 1 6. What is m? 25. wo parallel lines q and s are cut by a transversal t. 1 and 2 are a pair of alternate interior angles and m and 3 are vertical angles. What is m3? 26. has vertices (1, 9), (4, 3), and (x, 6). For what value of x is a right triangle with right? ixed eview For xercises 27 and 28, tell the postulate or theorem that you can use to prove the triangles congruent. ee esson Get eady! o prepare for esson 4-5, do xercises What is the side opposite? x 30. What is the angle opposite side? ee esson What is the angle opposite side? 32. Find the value of x hapter 4 ongruent riangles
Problem 2. Got It? Proving Triangle Parts Congruent to Measure Distance. Proof
4-4 Using orresponding arts of ongruent riangles ommon ore tate tandards G-..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. lso G-..12 1, 3 bjective
4-3. Triangle Congruence by ASA and AAS. Content Standard. Essential Understanding You can prove that two triangles are congruent
4-3 riangle ongruence by and ontent tandard G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. bjective o prove two triangles congruent using
Objectives To use relationships among sides and angles of parallelograms To use relationships among diagonals of parallelograms
6-2 roperties of arallelograms ontent tandards.o.11 rove theorems about parallelogram. s include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each
Objectives To use the AA Postulate and the SAS and SSS Theorems To use similarity to find indirect measurements
7-3 roving riangles imilar ontent tandards G..5 Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.G.5 rove the slope criteria for parallel and
EXERCISES Practice and Problem Solving
XI ractice and roblem olving or more practice, see xtra ractice. ractice by xample xample (page 224) In each diagram, the red and blue triangles are congruent. Identify their common side or angle.. K 2.
Proving Congruence ASA, AAS
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To prove two triangles congruent using the SSS and SAS Postulates. Are the triangles below congruent? How do you know? 6 B 4
4-2 riangle ongruence by SSS and SS ommon ore State Standards -SR..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. P 1, P 3, P 4, P 7 Objective
To identify congruence transformations To prove triangle congruence using isometries
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The hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. THEOREM 5.2. right triangles, and
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Proof EXAMPLE EXAMPLE. Given:
4-7 hat ou ll earn o identify congruent overlapping triangles o prove two triangles congruent by first proving two other triangles congruent... nd hy o identify overlapping triangles in scaffolding, as
To identify congruence transformations To prove triangle congruence using isometries
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5.3 Proving Triangles are
0 1 1 10 5.3 roving riangles are ongruent: and Goal how triangles are congruent using and. ey Words vertical angles p. 75 alternate interior angles p. 121 Geo-ctivity 1 raw a segment 3 inches long. abel
EXERCISES Practice and Problem Solving
XI ractice and roblem olving For more practice, see xtra ractice. ractice by xample lgebra Find the value of x in each parallelogram. xample (page 95. 5.. 0. 56 5. 80 6. 6 xample (page 95 lgebra Find the
Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.
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5 hapter eview 5.1 ngles of riangles (pp. 231 238) ynamic Solutions available at igideasath.com lassify the triangle by its sides and by measuring its angles. he triangle does not have any congruent sides,
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To recognize congruent figures and their corresponding parts
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Proving That a Quadrilateral Is a Parallelogram. To determine whether a quadrilateral is a parallelogram
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Essential Question How can you use congruent triangles to make an indirect measurement?
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