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G-09 Congruent Triangles and their parts
“I can name corresponding sides and angles of two triangles.”
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Reflexive Property AB = AB
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Symmetric Property Transitive Property
If A = B, then B = A If A = B and B = C, then A = C Transitive Property
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Addition, Subtraction, Multiplication, Division Property (=)
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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
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Angle/Segment Addition Postulate
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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.
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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
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Definition of Perpendicular Lines
Perpendicular lines intersect to form 90 angles. Perpendicular lines are form congruent angles
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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°
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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.
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Definition of Right Angles
All right angles are congruent If A and B are right angles, then A B
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Third Angle Theorem
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Definition of Congruent Triangles
If two or more triangles have corresponding angles and sides that are congruent, then those triangles are congruent.
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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
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Example 1 A. Given: ∆PQR ∆STU
Identify all pairs of corresponding congruent parts. Angles: Sides:
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Example 1 B. Given: ∆ABC ∆DEF
Identify all pairs of corresponding congruent parts. Angles: Sides:
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Example 1 C. Given: ∆JKM ∆LKM
Identify all pairs of corresponding congruent parts. Angles: Sides:
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Example 2 A. Given: polygon ABCD EFGH
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Example 2 B. Given: polygon ABCD EFGH
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Example 2 C. Given: polygon DEFGH IJKLM
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Example 3a: Given: K is the midpt. of JL, Prove: Statement Reason
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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
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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
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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
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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
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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
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Example 3d Given: PR and QT bisect each other. PQS RTS, QP RT
Prove: ∆QPS ∆TRS
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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
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Example 3e Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML. JK || ML. Prove: ∆JKN ∆LMN
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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
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Example 4a Given: ∆ABC ∆DBC. Find the value of x. Find mDBC.
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Example 4b Given: ∆ABC ∆DEF 1. Find the value of x. 2. Find mF.
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Example 4c Given: ∆ABD ∆CBD 1. Find the value of x. 2. Find AD.
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Example 4d Given: ∆RSU ∆TSU 1. Find the value of x. 2. Find UT.
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