Question

find the missing angle (triangle with 42° and a°, options: 90, 96, 138, 42)

Explanation:

Step1: Identify triangle type (isosceles)

The triangle has two equal sides (marked with ticks), so it's isosceles with two equal angles. One angle is \(42^\circ\), so another angle is also \(42^\circ\)? Wait, no—wait, the right angle? Wait, no, the triangle has a right angle? Wait, no, the angle marked \(a^\circ\) and the \(42^\circ\) angle—wait, no, the sum of angles in a triangle is \(180^\circ\). Wait, the triangle has two equal sides, so the base angles are equal? Wait, no, the angle with \(42^\circ\) and the other angle—wait, maybe it's a triangle with two equal sides, so two angles are equal. Wait, let's check: sum of angles in a triangle is \(180^\circ\). Let's denote the angles: \(42^\circ\), \(a^\circ\), and the third angle. Wait, no, the triangle has two equal sides, so the angles opposite them are equal. Wait, the two sides with ticks are equal, so the angles opposite them are equal. So one angle is \(42^\circ\), so the other equal angle? Wait, no, maybe I misread. Wait, the triangle has a \(42^\circ\) angle, and two sides with ticks, so the angles opposite those sides are equal. Wait, maybe the right angle? No, the angle marked \(a^\circ\) looks like a right angle? No, wait, let's calculate. Sum of angles in a triangle is \(180^\circ\). If two angles are equal (since two sides are equal), let's say the two equal angles are \(x\) and \(42^\circ\)? No, wait, maybe the triangle is isosceles with two angles equal, and one angle is \(42^\circ\), so the other two angles: wait, no, the sum is \(180\). Wait, maybe the triangle has a right angle? No, the angle marked \(a^\circ\) is not a right angle. Wait, let's do the math: sum of angles in a triangle is \(180^\circ\). So if one angle is \(42^\circ\), and the other two angles are equal (because two sides are equal), then let the equal angles be \(x\). So \(42 + x + x = 180\). Then \(2x = 180 - 42 = 138\), so \(x = 69\)? Wait, no, that's not one of the options. Wait, maybe I made a mistake. Wait, the options are 90, 96, 138, 42. Wait, maybe the triangle is a right triangle? No, the angle marked \(a^\circ\) is a right angle? Wait, no, the sum of angles: if one angle is \(42^\circ\), and another is \(90^\circ\), then the third is \(48^\circ\), but that's not an option. Wait, maybe the triangle has two angles equal, and one angle is \(42^\circ\), so the other angle is \(180 - 42 - 42 = 96^\circ\). Ah! That makes sense. So the two equal sides have opposite angles equal, so the two angles are \(42^\circ\) each, so the third angle is \(180 - 42 - 42 = 96^\circ\). So \(a = 96\).

Step2: Calculate the missing angle

Sum of angles in a triangle: \(180^\circ\). Two equal angles: \(42^\circ\) each.
\(a = 180 - 42 - 42 = 96\).

Answer:

96