Question

consider the two triangles shown below,

(images of two triangles: first with angles 63°, 72° and side 3; second with angles 45°, 63° and side 3; note: triangles not drawn to scale)

are the two triangles congruent?
choose 1 answer:
a yes
b no
c there is not enough information to say.

Explanation:

Step1: Find the third angle of each triangle

For the first triangle, the sum of angles in a triangle is \(180^\circ\). So the third angle is \(180 - 63 - 72 = 45^\circ\).
For the second triangle, the third angle is \(180 - 45 - 63 = 72^\circ\).

Step2: Check congruence conditions

Now, both triangles have angles \(45^\circ\), \(63^\circ\), \(72^\circ\) and a side of length 3. By the Angle - Side - Angle (ASA) or Angle - Angle - Side (AAS) congruence criteria, if two angles and a corresponding side are equal, the triangles are congruent. Here, we have two angles and a side (the side of length 3 is between the \(63^\circ\) angle and the other angle in each triangle? Wait, no, let's check the correspondence. Wait, in the first triangle, angles are \(63^\circ\), \(72^\circ\), \(45^\circ\) (we found the third angle as \(45^\circ\)) and a side of length 3 adjacent to \(63^\circ\) and \(72^\circ\)? Wait, no, the first triangle has angles \(63^\circ\), \(72^\circ\) and the side of length 3 is between \(63^\circ\) and \(72^\circ\)? Wait, the second triangle has angles \(45^\circ\), \(63^\circ\) and the side of length 3 is between \(63^\circ\) and the unknown angle (which we found as \(72^\circ\)). Wait, actually, let's list the angles:

First triangle angles: \(63^\circ\), \(72^\circ\), \(45^\circ\) (since \(63 + 72+45 = 180\)) and a side of length 3.

Second triangle angles: \(45^\circ\), \(63^\circ\), \(72^\circ\) (since \(45 + 63+72 = 180\)) and a side of length 3.

So, we can match the angles: \(63^\circ\) is common, one triangle has \(72^\circ\) and \(45^\circ\), the other has \(45^\circ\) and \(72^\circ\), and the side of length 3 is included between two angles. So by ASA (Angle - Side - Angle), the two triangles are congruent because we have two angles and the included side equal.

Answer:

A. Yes