Question

what is the value of t?

Explanation:

Step1: Identify congruent triangles

Triangles \(BCD\) and \(BAD\) are right triangles (since \(\angle C\) and \(\angle A\) are right angles) with \(BC = AD = 20\) (wait, no, \(CD = 20\) and \(BD\) is common? Wait, actually, \(CD = 20\), \(AD = 20\), and \(BD\) is common. So by HL (Hypotenuse-Leg) theorem, \(\triangle BCD \cong \triangle BAD\). Therefore, the angles at \(B\) should be equal? Wait, the angles at \(B\): \(\angle CBD = 2t\) and \(\angle ABD = t + 24^\circ\). Since the triangles are congruent, these angles should be equal. So set \(2t = t + 24^\circ\).

Step2: Solve for \(t\)

Subtract \(t\) from both sides: \(2t - t = t + 24^\circ - t\)
Which simplifies to \(t = 24^\circ\)? Wait, no, wait. Wait, maybe the angles are complementary? Wait, no, let's re-examine. Wait, the figure: \(BC\) and \(AD\) are both perpendicular to their respective axes, \(CD = 20\), \(AD = 20\), so \(CD = AD\), and \(BD\) is common. So right triangles \(BCD\) (right at \(C\)) and \(BAD\) (right at \(A\)) have \(CD = AD = 20\) and \(BD = BD\), so HL congruence. Therefore, \(\angle CBD = \angle ABD\)? Wait, no, \(\angle CBD\) is \(2t\), \(\angle ABD\) is \(t + 24\). So if the triangles are congruent, then \(\angle CBD = \angle ABD\), so \(2t = t + 24\). Then solving: \(2t - t = 24\), so \(t = 24\). Wait, but let's check again.

Wait, maybe the angles at \(B\) are equal because the triangles are congruent. So yes, \(2t = t + 24\), so \(t = 24\).

Answer:

\(24\)