Question

prove the following statement using the given pieces of information: given: ∠qhl ≅ ∠dhl, hl ⊥ gd prove: △hlp ≅ △hld. complete the two - column proof by filling in the blanks. statements reasons 1. h l ⊥ g d 1. given 2. ∠phl and ∠dhl are right angles. 2.

reasons options: definition of ⊥, vertical angles theorem, alternate interior angles converse theorem, right angles congruence theorem

Explanation:

Step1: Analyze the Given Information

We are given that \(\angle QHA=\angle DHA\) (right angles, since \(AH\perp QD\) as per the diagram's right angle marking) and \(HA = HA\) (common side). Also, we need to prove \(\triangle AHQ\cong\triangle AHD\).

Step2: Identify the Congruence Criterion

We have two right angles (\(\angle QHA\) and \(\angle DHA\)) and a common side \(HA\). Also, if we consider the hypotenuse - leg or angle - side - angle, but here since we have right angles, the Hypotenuse - Leg (HL) or more appropriately, the Angle - Side - Angle (ASA) or Side - Angle - Side (SAS) can be used. But in the context of right triangles, the Right Angle Congruence Theorem (which states that all right angles are congruent) and then the Hypotenuse - Leg or other congruence theorems. Wait, the given reasons include "Right Angles Congruence Theorem" which says that all right angles are congruent, so \(\angle QHA\cong\angle DHA\) by Right Angles Congruence Theorem. Then, \(HA\) is common, and if we consider the triangles \(\triangle AHQ\) and \(\triangle AHD\), we can use the Angle - Side - Angle (ASA) or Side - Angle - Side (SAS). But the reason for the step where we have \(\angle QHA\) and \(\angle DHA\) as right angles, the reason is "Right Angles Congruence Theorem" because all right angles are congruent.

Step3: Complete the Proof

1. Statement 1: \(\angle QHA\) and \(\angle DHA\) are right angles.

• Reason 1: Given (from the diagram, \(AH\perp QD\)) or by definition of perpendicular lines (if we consider the right angle marking). But the available reason is "Right Angles Congruence Theorem" is about congruence of right angles. Wait, the first blank for the reason when the statement is "1. \(\angle QHA\) and \(\angle DHA\) are right angles" - the reason should be "Definition of Perpendicular Lines" (since if two lines are perpendicular, they form right angles), but among the given options, the "Right Angles Congruence Theorem" is about congruence of right angles. Wait, maybe the first step:
• Statement: \(\angle QHA\cong\angle DHA\)
• Reason: Right Angles Congruence Theorem (because all right angles are congruent)
• Then, \(HA = HA\) (Reflexive Property, but not in the options). Wait, the options are "Definition of \(\perp\)", "Vertical Angles Theorem", "Alternate Interior Angles Converse Theorem", "Right Angles Congruence Theorem".
• The correct reason for \(\angle QHA\cong\angle DHA\) is "Right Angles Congruence Theorem" because all right angles are congruent.

Then, to prove \(\triangle AHQ\cong\triangle AHD\), we can use SAS (if we have another side) or ASA. But given the options, the "Right Angles Congruence Theorem" is used to show that the right angles are congruent.

Answer:

The reason for the statement that \(\angle QHA\) and \(\angle DHA\) are congruent (since they are right angles) is "Right Angles Congruence Theorem". So the answer is to fill the blank with "Right Angles Congruence Theorem" for the reason when the statement is about \(\angle QHA\) and \(\angle DHA\) being congruent (as right angles).