Question

select all the true statements for the figures shown. a. if ( mangle d + mangle e = 90 ), then ( mangle f = 90 ). the triangles are congruent by hl. b. if ( mangle d = 37 ), then ( mangle a = 37 ). the triangles are congruent by aas. c. if ( angle econgangle b ), then the triangles are congruent by sas. d. if ( angle f ) is a right angle, then the triangles are congruent by hl. e. if ( mangle d + mangle e = 90 ), then ( mangle f = 90 ). the triangles are congruent by ssa

Explanation:

Step1: Use triangle angle - sum property

In \(\triangle DEF\), by the triangle angle - sum property \(m\angle D + m\angle E+m\angle F=180^{\circ}\). If \(m\angle D + m\angle E = 90^{\circ}\), then \(m\angle F=180-(m\angle D + m\angle E)=90^{\circ}\). For right - triangle congruence (HL - Hypotenuse - Leg), if \(\triangle ABC\) and \(\triangle DEF\) are right - triangles (\(\angle C = 90^{\circ}\), and if \(\angle F = 90^{\circ}\)) and the hypotenuse and a leg are equal (by the markings on the sides), then \(\triangle ABC\cong\triangle DEF\) by HL. So, statement A is True.

Step2: Check AAS congruence

AAS (Angle - Angle - Side) requires two angles and a non - included side. We don't have enough information about the side correspondence for AAS if we just know \(m\angle D=m\angle A = 37^{\circ}\). The side markings do not support AAS. So, statement B is False.

Step3: Check SAS congruence

SAS (Side - Angle - Side) requires two sides and the included angle. The side markings and the given \(\angle E\cong\angle B\) do not give the included angle for the marked sides. So, statement C is False.

Step4: Check HL congruence

If \(\angle F\) is a right angle (\(\angle F = 90^{\circ}\)), then \(\triangle ABC\) (\(\angle C=90^{\circ}\)) and \(\triangle DEF\) are right - triangles. With the hypotenuse and a leg equal (by side markings), \(\triangle ABC\cong\triangle DEF\) by HL. So, statement D is True.

Step5: Check SSA congruence

SSA (Side - Side - Angle) is not a valid congruence criterion (except for the HL case in right - triangles). So, statement E is False.

Answer:

A. If \(m\angle D + m\angle E = 90\), then \(m\angle F = 90\). The triangles are congruent by HL.
D. If \(\angle F\) is a right angle, then the triangles are congruent by HL.