Question

18. YQ and XP are altitudes to the congruent sides of Isosceles triangle WXY. Keisha is going to prove that YQ S XP, by showing that they are congruent parts of congruent triangles QXY and PYX . tto A. AAS, because AWXY is isosceles, its base angles are congruent. Perpendicular lines form right angles, which are congruent; and segment XY is shared. B. SSS, because segment QP would be parallel to XY. C. SSA, because segment XY would be shared; segments XP and YQ are altitudes, and AWXY is isosceles, so base angles are congruent. D. ASA, because AWXY is isosceles, its base angles are congruent. Segment XY is shared; perpendicular lines form right angles, which are congruent.

Answer 1

A prove Keisha can use to show congruent parts of congruent triangles QXY and PYX is: A. AAS, because ΔWXY is isosceles, its base angles are congruent. Perpendicular lines form right angles, which are congruent; and segment XY is shared.

In Mathematics and Euclidean Geometry, AAS is an abbreviation for Angle-Angle-Side and it states that when two (2) angles and the non-included side (adjacent to only one of the angles) in two triangles are all equal, then the triangles are said to be congruent.

Based on the angle, angle, side (AAS) congruence theorem, we can logically deduce that triangle QXY and triangle PYX are both congruent based on the following statements;

YQ = XP

∠WXY = ∠WYX

WQ/QX = WP/PY

PQ ║ XY

∠PXY = ∠QYX

The SSS congruence theorem cannot be used because we can't tell if sides XQ and PY or sides QY and PX are congruent. SSA is not a valid congruence theorem, so it cannot be used.

Since we can't tell if angles QYX and PXY are congruent, the ASA congruence theorem cannot be used.

Answer 2

Consider the given figure.

It is required to prove that traingles PYX and QXY are congruent.

Both the triangles share a common base XY.

The lines PX and QY are perpendicular on congruent sides, so PX must be equal to QY.

According to the theorem "Angles opposite to equal sides are equal", it follows that anglea PYX and QXY are equal.

Thus, it is observed that the two triangles PYX and QXY have 2 sides and one angle equal. Therefore, the two triangles will be congruent on the basis of SSA criteria.

Therefore, option C is the correct choice.