Question

Which triangles in the diagram are congruent?

Set of 4 triangles are plotted on a grid. Triangle 1 has 2 congruent sides and the angle between them is 70 degrees. Triangle 2 has 3 congruent sides. Triangle 3 has 2 congruent sides and the angle between them is 70 degrees.
A.
triangle 1 and triangle 2
B.
triangle 1, triangle 2, and triangle 3
C.
triangle 1 and triangle 3
D.
triangle 1, triangle 3, and triangle 4

Answer 1

Answer: triangle 1, triangle 2, and triangle 3

Step-by-step explanation:

Triangles 1 and 3 are congruent by SAS. So, by CPCTC, all three sides are congruent, making triangle 2 congruent to them as well by SSS.

Answer 2

Answer:

  C.  Δ1 ≅ Δ3

Step-by-step explanation:

There are five (5) congruence postulates that tell the ways triangles can be proven congruent. Their abbreviations are ...

  SSS, SAS, ASA, AAS, HL

What is congruence?

Two plane figures are congruent if all corresponding sides are identical in length (congruent), and all corresponding angles are identical in measure (congruent). Showing triangle congruence is simplified somewhat by the requirement that the sum of angles is always 180°. That means any two angles are sufficient to define the third.

In the above list of congruence postulates, S represents corresponding sides, and A represents corresponding angles. The sequence of S and A represents the sequence of the corresponding parts in the triangle. You note there is no SSA postulate, because there are cases where these measures in this order give rise to two different (not congruent) triangles. The HL postulate (hypotenuse-leg) is an exception, where the angle is a right angle.

What measures are given?

In the given figures, we can use the double hash marks on the line segments to mean this is a side that is congruent to a corresponding side (somewhere). Similarly, we can use the angle arc and angle measure to mean this angle is congruent to one with the same measure (somewhere). Then the congruence postulates we need to use with the different triangles are ...

1. SAS
2. SSS
3. SAS
4. SSA (no such postulate)

The congruent triangles

We notice the only two triangles that have corresponding sets of given values are Triangle 1 (SAS) and Triangle 3 (SAS).

That is, triangles 1 and 3 are congruent by the SAS postulate.

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Additional comment

The unfortunate case of non-unique triangles arises when SSA has the angle opposite the shorter side (the first S). Only the HL case can guarantee the angle is opposite the longest side (90° is opposite H). Even if the angle is somehow known to be opposite the longer side, there is no SSA postulate that can be invoked to demonstrate congruence.

The isosceles triangle of Triangle 4 could be shown to be congruent to one similarly marked, because one angle in an isosceles triangle is sufficient to define all three. However, in this problem statement, there is no triangle similarly marked.