Question

which triangles in the diagram are congruent?
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

Explanation:

Step1: Recall the SAS (Side - Angle - Side) congruence criterion

Two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.

Step2: Analyze triangle 1 and triangle 4

• For triangle 1: Let the equal - marked sides be \(a\) and \(b\), and the given angle (\(70^{\circ}\)) is between the two equal - marked sides.
• For triangle 4: The two equal - marked sides (same length as in triangle 1) and the \(70^{\circ}\) angle is also between the two equal - marked sides.

Step3: Analyze triangle 3

• In triangle 3, using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), if two sides are equal (isosceles triangle), and the vertex angle is \(70^{\circ}\), then the base angles are \(\frac{180 - 70}{2}=55^{\circ}\). But this does not match the side - angle - side configuration of triangle 1.

Step4: Analyze triangle 2

• Triangle 2 has two equal sides, but the angle between them is not \(70^{\circ}\) (by visual inspection of side - angle - side correspondence with triangle 1).

Answer:

D. triangle 1, triangle 3, and triangle 4