G-09 Congruent Triangles and their parts “I can name corresponding sides and angles of two triangles.”
Reflexive Property AB = AB
Symmetric Property Transitive Property If A = B, then B = A If A = B and B = C, then A = C Transitive Property
Addition, Subtraction, Multiplication, Division Property (=)
Distributive Property If A(B + C), then AB + AC Or If (B + C)A, then BA + CA If A = B, then A can be substituted for any B in the expression Substitution
Angle/Segment Addition Postulate
Definition of Congruence If AB = CD, then AB CD Congruent segments are segments that have the same length. Congruent angles are angles that have the same measure.
Definition of Vertical Angles Vertical angles are two nonadjacent angles formed by two intersecting lines. Vertical Angles are congruent 1 and 2 are vertical angles
Definition of Perpendicular Lines Perpendicular lines intersect to form 90 angles. Perpendicular lines are form congruent angles
Definition of Complementary/Supplementary Angles Complementary Angles: 2 angles that add up to be 90° Supplementary Angles: 2 angles that add up to be 90°
Definition of Midpoint/Bisector The midpoint M of AB is the pt that bisects, or divides, the segment into 2 congruent segments. (segments) If M is the midpt of AB, then AM = MB An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM.
Definition of Right Angles All right angles are congruent If A and B are right angles, then A B
Third Angle Theorem
Definition of Congruent Triangles If two or more triangles have corresponding angles and sides that are congruent, then those triangles are congruent.
In a congruence statement, the order of the vertices indicates the corresponding parts. When you write a statement such as ABC DEF, you are also stating which parts are congruent. Helpful Hint
Example 1 A. Given: ∆PQR ∆STU Identify all pairs of corresponding congruent parts. Angles: Sides:
Example 1 B. Given: ∆ABC ∆DEF Identify all pairs of corresponding congruent parts. Angles: Sides:
Example 1 C. Given: ∆JKM ∆LKM Identify all pairs of corresponding congruent parts. Angles: Sides:
Example 2 A. Given: polygon ABCD EFGH
Example 2 B. Given: polygon ABCD EFGH
Example 2 C. Given: polygon DEFGH IJKLM
Example 3a: Given: K is the midpt. of JL, Prove: Statement Reason
Statement Reason K is the midpt. of Given Definition of Midpoint Reflexive Property are right angles Definition of Perpendicular lines Right angles are congruent Third Angle Thm. Definition of Congruent Triangles
Given: YWX and YWZ are right angles. Example 3b Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY YZ. Prove: ∆XYW ∆ZYW
Example 3b: Statement Reason YWX and YWZ are right angles. Given YWX YWZ YW bisects XYZ XYW ZYW W is mdpt. of XZ XW ZW YW YW X Z XY YZ ∆XYW ∆ZYW
Given: AD bisects BE. BE bisects AD. AB DE, A D Example 3c Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ∆ABC ∆DEC
Example 3c: Statement Reason A D Given BCA DCE ABC DEC AB DE AD bisects BE, BE bisects AD BC EC, AC DC ∆ABC ∆DEC
Example 3d Given: PR and QT bisect each other. PQS RTS, QP RT Prove: ∆QPS ∆TRS
Example 3d: Statement Reason QP RT Given PQS RTS PR and QT bisect each other QS TS, PS RS QSP TSR QSP TRS ∆QPS ∆TRS
Example 3e Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML. JK || ML. Prove: ∆JKN ∆LMN
Example 3e: Statement Reason JK ML Given JK || ML JKN NML JL and MK bisect each other. JN LN, MN KN Vert. s Thm. Third s Thm. ∆JKN ∆LMN Def. of ∆s
Example 4a Given: ∆ABC ∆DBC. Find the value of x. Find mDBC.
Example 4b Given: ∆ABC ∆DEF 1. Find the value of x. 2. Find mF.
Example 4c Given: ∆ABD ∆CBD 1. Find the value of x. 2. Find AD.
Example 4d Given: ∆RSU ∆TSU 1. Find the value of x. 2. Find UT.