Question

directions: decide whether enough information is given to prove that that the triangles are congruent using the sss, sas, asa, aas, or hl congruence theorem.
1.
2.
3.
4.
5.
6.
7.
8.

Explanation:

Step1: Analyze Problem 1

The first figure has two triangles sharing a common side, with three pairs of congruent sides (marked by tick marks). So, by SSS (Side - Side - Side) congruence theorem, the triangles are congruent.

Step2: Analyze Problem 2

Both are right - triangles. One leg and the hypotenuse of each triangle are marked congruent (tick marks). By HL (Hypotenuse - Leg) congruence theorem for right - triangles, the triangles are congruent.

Step3: Analyze Problem 3

The figure is a square (implies all sides equal) with a diagonal. The two triangles are right - triangles, with legs equal (sides of square) and hypotenuse (diagonal) common. By HL or SSS, the triangles are congruent.

Step4: Analyze Problem 4

We have vertical angles (equal) and two pairs of congruent sides (tick marks) and a pair of equal angles. By ASA (Angle - Side - Angle) or SAS (Side - Angle - Side), the triangles are congruent (vertical angles are equal, and sides with tick marks and included angle).

Step5: Analyze Problem 5

We have two angles and a side (AAS - Angle - Angle - Side) congruent (marked angles and side with tick mark). So, by AAS congruence theorem, the triangles are congruent.

Step6: Analyze Problem 6

We have vertical angles (equal), a pair of equal angles (marked) and a side? Wait, looking at the figure, we have two triangles with vertical angles, a pair of equal angles and sides? Wait, actually, we have two angles and a side? Wait, the figure shows vertical angles (equal), a pair of marked angles and sides? Wait, more accurately, by ASA or AAS, since we have vertical angles (equal), a pair of equal angles (marked) and the included side? Wait, the triangles have vertical angles, a pair of equal angles and a side. So, by AAS or ASA, the triangles are congruent.

Step7: Analyze Problem 7

We have two sides (tick marks) and the included angle (marked) congruent. By SAS (Side - Angle - Side) congruence theorem, the triangles are congruent.

Step8: Analyze Problem 8

We have two sides (tick marks) and the included angle (marked) congruent. By SAS (Side - Angle - Side) congruence theorem, the triangles are congruent.

Answer:

1. SSS (Triangles are congruent by SSS)
2. HL (Triangles are congruent by HL)
3. HL (or SSS, Triangles are congruent by HL)
4. ASA (or SAS, Triangles are congruent by ASA)
5. AAS (Triangles are congruent by AAS)
6. ASA (or AAS, Triangles are congruent by ASA)
7. SAS (Triangles are congruent by SAS)
8. SAS (Triangles are congruent by SAS)