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Identify Triangle Congruence Theorems and Write Congruence Statements
Mathematics (Geometry)
Grade 9 (Junior High School)
Question Content
State the theorem that proves the triangles are congruent. Then, write a congruence statement for each of the 4 sets of triangles: \nA) Right triangles △AEB and △DCB with right angles at A and C, AB=BC, shared segment EB=BD? No, correction: Right angles at E and D, AE=DC, AB=BC\nB) Quadrilateral with triangles △MHT and △MAT, right angle at T, MT shared, MH=MA\nC) Triangles △MIN and △PIN with MI=PI, MN=PN, IN shared\nD) Triangles △ABC and △EDF with vertical angles at the intersection, AB=ED, BC=DF
Correct Answer
A) Theorem: HL (Hypotenuse-Leg) Congruence; Congruence Statement: △AEB ≅ △CDB\nB) Theorem: HL (Hypotenuse-Leg) Congruence; Congruence Statement: △MHT ≅ △MAT\nC) Theorem: SSS (Side-Side-Side) Congruence; Congruence Statement: △MIN ≅ △PIN\nD) Theorem: SAS (Side-Angle-Side) Congruence; Congruence Statement: △ABC ≅ △EDF
Detailed Solution Steps
1
Step 1: Analyze Figure A: Both are right triangles (∠E and ∠D are right angles). Hypotenuses AB=BC, legs AE=CD. This matches the HL congruence theorem for right triangles. Match corresponding vertices: A↔C, E↔D, B↔B to write the congruence statement.
2
Step 2: Analyze Figure B: Both are right triangles (∠HTM and ∠ATM are right angles). Hypotenuses MH=MA, leg MT is shared (so MT=MT). This fits the HL theorem. Match corresponding vertices: M↔M, H↔A, T↔T for the congruence statement.
3
Step 3: Analyze Figure C: All three pairs of corresponding sides are equal: MI=PI, MN=PN, IN=IN (shared side). This matches the SSS congruence theorem. Match corresponding vertices: M↔P, I↔I, N↔N for the congruence statement.
4
Step 4: Analyze Figure D: Vertical angles at the intersection are equal (∠ABC=∠EDF). The sides forming these angles are equal: AB=ED, BC=DF. This fits the SAS congruence theorem. Match corresponding vertices: A↔E, B↔D, C↔F for the congruence statement.
Knowledge Points Involved
1
HL (Hypotenuse-Leg) Congruence Theorem
This theorem applies only to right triangles. If the hypotenuse and one leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, the two triangles are congruent. It is a special case because right triangles have a built-in right angle that does not need to be explicitly proven equal.
2
SSS (Side-Side-Side) Congruence Theorem
If all three pairs of corresponding sides of two triangles are congruent, then the two triangles are congruent. This works because three fixed side lengths uniquely define the shape and size of a triangle (triangle rigidity).
3
SAS (Side-Angle-Side) Congruence Theorem
If two pairs of corresponding sides of two triangles are congruent, and the included angle (the angle between the two sides) is also congruent, then the two triangles are congruent. The angle must be the included angle, not any angle, to guarantee congruence.
4
Vertical Angles Theorem
When two lines intersect, the opposite (vertical) angles formed are congruent. This is used to identify equal angles in intersecting figure problems, often to satisfy the angle condition in congruence theorems like SAS.
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