Question

lesson 4 - 4/4 - 5: exit ticket
for the figures shown, which information can be used to show that \\( \triangle abc \cong \triangle def \\)? select all that apply
\\( m\angle e = 90 ^ { \circ } \\)
\\( m\angle d = 37 ^ { \circ } \\)
\\( \angle e \cong \angle b \\)
\\( \angle f \\) is a right angle

Explanation:

Step1: Recall congruence criteria for right - angled triangles

For right - angled triangles, we can use the Hypotenuse - Leg (HL) criterion. If we know that two right - angled triangles have a congruent hypotenuse and a congruent leg, they are congruent.

Step2: Analyze each option

• Option \(m\angle E = 90^{\circ}\):

If \(m\angle E=90^{\circ}\), then \(\triangle ABC\) and \(\triangle DEF\) are both right - angled triangles. Given the marked equal sides (one leg and the hypotenuse can be considered if we have right angles), we can potentially use the HL criterion.

• Option \(m\angle D = 37^{\circ}\):

Just knowing the measure of \(\angle D\) does not help in establishing congruence using any of the triangle congruence criteria (SSS, SAS, ASA, AAS, HL).

• Option \(\angle E\cong\angle B\):

If \(\angle E\cong\angle B\), along with the marked equal sides, we can use the AAS (Angle - Angle - Side) criterion.

• Option \(\angle F\) is a right angle:

If \(\angle F\) is a right angle, then \(\triangle ABC\) and \(\triangle DEF\) are both right - angled triangles. Given the marked equal sides (one leg and the hypotenuse can be considered if we have right angles), we can potentially use the HL criterion.

Answer:

\(m\angle E = 90^{\circ}\), \(\angle E\cong\angle B\), \(\angle F\) is a right angle.