4.2 Congruence & Triangles Geometry

Objectives: Identify congruent figures and corresponding parts Prove that two triangles are congruent

Identifying congruent figures Two geometric figures are congruent if they have exactly the same size and shape. NOT CONGRUENT CONGRUENT

Congruency 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.

Triangles Corresponding angles Corresponding Sides A B C Q P R

How do you write a congruence statement? There is more than one way to write a congruence statement, but it is important to list the corresponding angles in the same order. Normally you would write ∆ABC ≅ ∆PQR, but you can also write that ∆BCA ≅ ∆QRP or ∆CAB ≅ ∆RPQ.

Ex. 1 Naming congruent parts These are congruent triangles. Write congruence statements to identify all congruent corresponding parts.

Ex. 1 Naming congruent parts The diagram indicates that ∆DEF ≅ ∆RST. The congruent angles and sides are as follows: Angles: Sides:

Ex. 2 Using properties of congruent figures In the diagram, NPLM ≅ EFGH Find the value of x. 8 m 110° 87° 10 m 72° (7y+9)° (2x - 3) m

Ex. 2 Using properties of congruent figures In the diagram NPLM ≅ EFGH Find the value of y. 8 m 110° 87° 10 m 72° (7y+9)° (2x - 3) m

Third Angles Theorem If any 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.

Ex. 3 Using the Third Angles Theorem 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°) = 180° – 120° = 60° Then, mM = mT 60° = (2x + 30)° 30 = 2x 15 = x Find the value of x. (2x + 30)° 55° 65°

Ex. 4 Proving Triangles are congruent Decide whether the triangles are congruent. Justify your reasoning. From the diagram, you are given that all three pairs of corresponding sides are congruent. 92° 92°

Ex. 4 Proving Triangles are congruent 92° 92°

Ex. 4 Proving Triangles are congruent So all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, ∆PQR ≅ ∆NQM. 92° 92°

Ex. 5 Proving two triangles are congruent

Ex. 5 Proving two triangles are congruent

Proof: Statements: Reasons:

Proof: Statements: Reasons: 1. Given

Proof: 1. Given 2. Alternate Interior Angles Theorem Statements: Reasons: 1. Given 2. Alternate Interior Angles Theorem

Proof: 1. Given 2. Alternate Interior Angles Theorem Statements: Reasons: 1. Given 2. Alternate Interior Angles Theorem 3. Vertical Angles Theorem

Proof: 1. Given 2. Alternate Interior Angles Theorem Statements: Reasons: 1. Given 2. Alternate Interior Angles Theorem 3. Vertical Angles Theorem 4. Given

Proof: 1. Given 2. Alternate Interior Angles Theorem Statements: Reasons: 1. Given 2. Alternate Interior Angles Theorem 3. Vertical Angles Theorem 4. Given 5. Definition of a Midpoint

Proof: 1. Given 2. Alternate Interior Angles Theorem Statements: Reasons: 1. Given 2. Alternate Interior Angles Theorem 3. Vertical Angles Theorem 4. Given 5. Definition of a Midpoint 6. Definition of Congruent Triangles

What should you have learned? To prove two triangles congruent by the definition of congruence—that is all pairs of corresponding angles and corresponding sides are congruent. In upcoming lessons you will learn more efficient ways of proving triangles are congruent. The properties on the next slide will be useful in such proofs.

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 ≅ ∆JKL.