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 e cong angle 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: Analyze option A

In triangle \(ABC\) and \(DEF\), if \(m\angle D + m\angle E=90\), we cannot directly conclude that \(m\angle F = 90\). Because in a triangle, the sum of interior angles is \(180\), so \(m\angle F=180-(m\angle D + m\angle E)\). If \(m\angle D + m\angle E = 90\), then \(m\angle F=90\). But for the HL (Hypotenuse - Leg) congruence criterion, we need to have right - angled triangles. However, we don't know if the sides (hypotenuse and leg) are equal. So, this statement is wrong.

Step2: Analyze option B

If \(m\angle D = 37\), we cannot say \(m\angle A=37\) just like that. There is no information given about the relationship between \(\angle A\) and \(\angle D\) in terms of congruence of triangles (no congruence criterion is satisfied with just one angle measure). So, this statement is wrong.

Step3: Analyze option C

For SAS (Side - Angle - Side) congruence criterion, we need two sides and the included angle to be equal. Here, we don't know if \(\angle E\) and \(\angle B\) are included angles between the corresponding equal sides. So, this statement is wrong.

Step4: Analyze option D

If \(\angle F\) is a right - angle (\(m\angle F = 90\)), then \(\triangle DEF\) is a right - angled triangle. In \(\triangle ABC\), \(\angle C = 90\). Also, from the figure, we can assume that the hypotenuse and one leg are equal (by the markings on the sides). So, by HL (Hypotenuse - Leg) congruence criterion (for right - angled triangles, if the hypotenuse and one leg of one right - angled triangle are equal to the hypotenuse and one leg of another right - angled triangle, then the two triangles are congruent), \(\triangle ABC\cong\triangle DEF\).

Step5: Analyze option E

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

Answer:

D. If \(\angle F\) is a right angle, then the triangles are congruent by HL.