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Recall Angle-Angle-Angle Similarity Theorem (AAA theorem), if two  pairs of corresponding angles between two triangles are respectively equal to each other then, the triangles are similar.

Since QR || YZ and line QZ (and YR) is the transversal that cuts the QR and YZ, the angles that QZ makes can remind us of the types of angles made a transversal.

Angle XZY and XQR are considered alternate interior angles since they're both "inside" the parallel lines QR and YZ but are oppositely located relative to each other (angle XZY is on the bottom right while angle XQR is on the top left) and to the transversal line. Alternate interior angles have the same angle measure so angle XZY = angle XQR.

Number of angles shared between both triangles: 1.

Since QZ and YR cross each other at point X, they create 2 pairs of vertical angles. Recall that vertical angles are angles that are oppositely located from each other at a cross section. So, angle QXR and angle ZXY are vertical angles. Vertical angles have the same angle measure so angle QXR and ZXY are equal to each other.

Number of angles shared between both triangles: 2.

Since triangle QXR and YXZ satisfy the AAA theorem, they can be deemed similar.

Answer:

see explanation

Step-by-step explanation:

There are 3 ways to prove triangles are similar

• AA : If two angles of one triangle are congruent (equal) to two angles of another triangle, then the triangles are similar.

• SSS : If the ratios of the three pairs of corresponding sides of two triangles are equal, then the triangles are similar.

• SAS : If the ratios of two pairs of corresponding sides of two triangles are equal and the included angles are congruent, then the triangles are similar.

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Given QR and YZ are parallel , then

∠ QRY = ∠ RYZ ( alternate angles )

∠ QXR = ∠ ZXY ( vertically opposite angles )

Then Δ  XYZ and Δ XRQ are similar , by the AA postulate

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