Question

what is the value of a?
(image of triangle xwv with x at top, w at bottom left, v at bottom right. sides xw and xv have one tick mark each, side wv has one tick mark. angle at v is 34 degrees, angle at x is a. a = blank °)

Explanation:

Step1: Identify Triangle Type

The triangle has two equal sides (marked with ticks), so it's isosceles. In an isosceles triangle, the angles opposite equal sides are equal. Here, sides \(XV\) and \(WV\) are equal? Wait, no—wait, the ticks on \(XV\) and \(WV\)? Wait, looking at the triangle: sides \(XV\) (from X to V) and \(WV\) (from W to V)? Wait, no, the ticks: one on \(XV\) and one on \(WV\)? Wait, no, the triangle is \(XWV\) with vertices X, W, V. The sides: from X to V (has a tick), from W to V (has a tick). So sides \(XV = WV\), so the angles opposite them: angle at W (opposite \(XV\)) and angle at X (opposite \(WV\))? Wait, no, angle at V is \(34^\circ\). Wait, maybe the two equal sides are \(XW\) and \(XV\)? Wait, no, the ticks: one on \(XV\) (the side from X to V) and one on \(WV\) (from W to V). Wait, maybe it's an isosceles triangle with \(XV = WV\), so the base is \(XW\), and the equal sides are \(XV\) and \(WV\). Then the base angles would be angle at X (a) and angle at W? Wait, no, angle at V is \(34^\circ\). Wait, maybe I made a mistake. Wait, in a triangle, the sum of angles is \(180^\circ\). If two sides are equal, the triangle is isosceles, so the base angles are equal. Wait, let's re-examine: the triangle has vertices X, W, V. The side from X to V has a tick, the side from W to V has a tick. So \(XV = WV\), so the angles opposite these sides: angle opposite \(XV\) is angle at W (angle W), angle opposite \(WV\) is angle at X (angle a). Wait, no, angle opposite \(XV\) is angle at W (since \(XV\) is between X and V, so opposite angle is W). Angle opposite \(WV\) is angle at X (since \(WV\) is between W and V, opposite angle is X). So if \(XV = WV\), then angle W = angle X (a). But angle at V is \(34^\circ\). Then sum of angles: angle X + angle W + angle V = \(180^\circ\). Since angle X = angle W = a, then \(a + a + 34^\circ = 180^\circ\). Wait, but that would be \(2a + 34 = 180\), so \(2a = 146\), \(a = 73\)? No, wait, maybe the equal sides are \(XW\) and \(XV\), so the base is \(WV\), and the equal angles are angle at V and angle at W? Wait, angle at V is \(34^\circ\), so angle at W is also \(34^\circ\), then angle at X (a) is \(180 - 34 - 34 = 112\)? No, that doesn't make sense. Wait, maybe the two equal sides are \(XW\) and \(XV\), so the base is \(WV\), and the equal angles are angle at V (34°) and angle at W. Wait, no, the ticks: one on \(XV\) (from X to V) and one on \(WV\) (from W to V). Wait, maybe the triangle is isosceles with \(XV = XW\), so the equal sides are \(XV\) and \(XW\), so the base is \(WV\), and the base angles are angle at V (34°) and angle at W. Then angle at W is 34°, so angle at X (a) is \(180 - 34 - 34 = 112\)? No, that can't be. Wait, maybe the two equal sides are \(XW\) and \(WV\), so the equal angles are angle at X (a) and angle at V (34°). Then angle at W is \(180 - a - 34\). But if \(XW = WV\), then angle at V (34°) = angle at X (a). Then a = 34°, but that would make it isoceles with two angles 34°, but then the third angle is 112°. But that contradicts the ticks. Wait, maybe I misread the ticks. Let me look again: the triangle has X at the top, W at the bottom left, V at the bottom right. The side from X to V (right side) has a tick, the side from W to V (bottom side) has a tick. So sides \(XV = WV\). So the triangle is isosceles with \(XV = WV\), so the base is \(XW\), and the equal sides are \(XV\) and \(WV\). Then the base angles are angle at X (a) and angle at W. The vertex angle is angle at V (34°). Wait, no, in an isosceles triangle, the vertex angle is the angle between the two equal sides. Wait, the two equal sides are \(XV\) and \(WV\), so the angle between them is angle at V (34°). Then the base angles (angle at X and angle at W) are equal. So sum of angles: angle X + angle W + angle V = 180. Since angle X = angle W = a, then \(2a + 34 = 180\). Solving for a: \(2a = 180 - 34 = 146\), so \(a = 146 / 2 = 73\). Wait, that makes sense. So angle at X (a) is 73 degrees.

Step2: Calculate Angle a

Sum of angles in a triangle: \(180^\circ\).
Equal sides \(XV = WV\) ⇒ angles opposite (\(∠X\) and \(∠W\)) are equal. Let \(∠X = ∠W = a\).
\(∠V = 34^\circ\).
So:
\(a + a + 34^\circ = 180^\circ\)
\(2a = 180^\circ - 34^\circ = 146^\circ\)
\(a = \frac{146^\circ}{2} = 73^\circ\)

Answer:

\(73\)