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 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 F 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
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 s 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
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. 1. 2. 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 2. 8. 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
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 j @ jq. 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? 15. Given:, bisects rove: #, bisects 16. Given: / #, / bisects at, is on / rove: = Q 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: 19. Given: } Q, Q rove: Q bisects. F Q esson 4-4 Using orresponding arts of ongruent riangles 247
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: # F pply What You ve earned ook back at the information on page 217 and at your work from the pply What You ve earned sections in essons 4-1 and 4-3. hoose from the following words and names of figures to complete the sentences below. HI I 7 congruent corresponding Y Y YX X YX YX YX In the pply What You ve earned in esson 4-3, you proved that a.?. ow, you can conclude that b.? because c.? parts of d.? triangles are congruent. imilarly, e.? and X are congruent corresponding sides. nother pair of congruent corresponding sides are f.? and YX. 248 hapter 4 ongruent riangles