Identifying Congruent Figures Two geometric figures are congruent

Identifying Congruent Figures Two geometric figures are congruent if they have exactly the same size and shape. Each of the red figures is congruent to the other red figures. None of the blue figures is congruent to another blue figure. Goal 1

Identifying Congruent Figures When two figures are congruent, there is a correspondence between their angles and sides such that corresponding angles are congruent and corresponding sides are congruent. For the triangles below, you can write ABC PQR , which reads “triangle ABC is congruent to triangle PQR. ” The notation shows the congruence and the correspondence. Corresponding Angles Corresponding Sides A P B Q C R AB PQ BC QR CA RP There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. For example, you can also write BCA QRP. Goal 1

Naming Congruent Parts The two triangles shown below are congruent. Write a congruence statement. Identify all pairs of congruent corresponding parts. SOLUTION The diagram indicates that DEF RST. The congruent angles and sides are as follows. Angles: D R, E S, F T Sides: DE RS , EF ST , FD TR Example

Using Properties of Congruent Figures In the diagram, NPLM EFGH. Find the value of x. SOLUTION You know that LM GH. So, LM = GH. 8 = 2 x – 3 11 = 2 x 5. 5 = x Example

Using Properties of Congruent Figures In the diagram, NPLM EFGH. Find the value of x. Find the value of y. SOLUTION You know that LM GH. You know that N E. So, LM = GH. So, m N = m E. 8 = 2 x – 3 11 = 2 x 5. 5 = x 72˚ = (7 y + 9)˚ 63 = 7 y 9=y Example

Identifying Congruent Figures The Third Angles Theorem below follows from the Triangle Sum Theorem. THEOREM Theorem 4. 3 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If A D and B E, then C F. Goal 1

Using the Third Angles Theorem Find the value of x. SOLUTION In the diagram, N R and L S. From the Third Angles Theorem, you know that M T. So, m M = m T. From the Triangle Sum Theorem, m M = 180˚– 55˚ – 65˚ = 60˚. m M = m T 60˚ = (2 x + 30)˚ Third Angles Theorem Substitute. 30 = 2 x Subtract 30 from each side. 15 = x Divide each side by 2. Example

Proving Triangles are Congruent Decide whether the triangles are congruent. Justify your reasoning. SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. RP MN, PQ NQ , and QR QM Because P and N have the same measures, P N. By the Vertical Angles Theorem, you know that PQR NQM. By the Third Angles Theorem, R M. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, PQR NQM. Goal 2

Proving Two Triangles are Congruent Prove that AEB DEC. A B E C GIVEN PROVE D AB || DC , AB DC, E is the midpoint of BC and AD. AEB DEC. Plan for Proof Use the fact that AEB and DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent. Example

Proving Two Triangles are Congruent AEB Prove that DEC. A B E SOLUTION D C Statements Reasons AB || DC , AB DC Given EAB EDC, ABE DCE Alternate Interior Angles Theorem AEB DEC Vertical Angles Theorem E is the midpoint of AD, E is the midpoint of BC Given AE DE , BE CE Definition of midpoint AEB DEC Definition of congruent triangles Example

Proving Triangles are Congruent In this lesson, you have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. In upcoming lessons, you will learn more efficient ways of proving that triangles are congruent. The properties below will be useful in such proofs. THEOREM B Theorem 4. 4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles If ABC DEF , then DEF ABC. Transitive Property of Congruent Triangles If ABC DEF and DEF JKL , then ABC A C E D F L JKL. J K Goal 2