Question

find the measure of the three missing angles in the parallelogram below. (with a parallelogram image labeled with angles: 103°, x°, y°, z°)

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)).

Step2: Find angle \(x\)

Since opposite angles in a parallelogram are equal, the angle opposite to \(103^\circ\) is equal to it? Wait, no, wait. Wait, the angle given is \(103^\circ\), and angle \(x\) is opposite to which angle? Wait, no, let's look at the diagram. The angle \(103^\circ\) and angle \(x\): Wait, no, in a parallelogram, consecutive angles are supplementary. Wait, the angle \(103^\circ\) and angle \(x\): Wait, no, the angle \(103^\circ\) and angle \(y\) are consecutive? Wait, no, let's label the parallelogram. Let's say the vertices are, in order, \(A(103^\circ)\), \(B(z^\circ)\), \(C(y^\circ)\), \(D(x^\circ)\), so \(AB\) is adjacent to \(BC\), so angle at \(A\) is \(103^\circ\), angle at \(B\) is \(z^\circ\), angle at \(C\) is \(y^\circ\), angle at \(D\) is \(x^\circ\). Then, in a parallelogram, angle \(A\) is equal to angle \(C\), and angle \(B\) is equal to angle \(D\). Also, angle \(A\) + angle \(B\) = \(180^\circ\) (consecutive angles are supplementary).

So angle \(A = 103^\circ\), so angle \(C = y = 103^\circ\)? Wait, no, that can't be. Wait, no, maybe I got it wrong. Wait, no, in a parallelogram, opposite angles are equal, so angle \(A\) (103°) is equal to angle \(C\) (y°), so \(y = 103^\circ\)? Wait, no, that would mean consecutive angles are equal, which would make it a rectangle, but that's not the case. Wait, no, I think I mixed up. Wait, no, consecutive angles are supplementary. So angle \(A\) (103°) and angle \(B\) (z°) are consecutive, so \(103 + z = 180\), so \(z = 180 - 103 = 77^\circ\). Then angle \(B\) (z°) is equal to angle \(D\) (x°), so \(x = z = 77^\circ\). And angle \(A\) (103°) is equal to angle \(C\) (y°), so \(y = 103^\circ\). Wait, that makes sense. Let's verify:

In a parallelogram, opposite angles are equal: so angle \(A\) (103°) = angle \(C\) (y°) ⇒ \(y = 103^\circ\).

Consecutive angles are supplementary: angle \(A\) (103°) + angle \(B\) (z°) = 180° ⇒ \(z = 180 - 103 = 77^\circ\).

Then angle \(B\) (z°) = angle \(D\) (x°) ⇒ \(x = 77^\circ\).

Wait, that makes sense. So:

• Angle \(x\): opposite to angle \(z\), so \(x = z\). Since angle \(103^\circ\) and angle \(z\) are consecutive, \(103 + z = 180\) ⇒ \(z = 77^\circ\), so \(x = 77^\circ\).

• Angle \(y\): opposite to angle \(103^\circ\), so \(y = 103^\circ\).

• Angle \(z\): consecutive to \(103^\circ\), so \(z = 180 - 103 = 77^\circ\).

Wait, let's check again:

1. Opposite angles are equal: so angle \(103^\circ\) (let's call it angle \(A\)) is equal to angle \(C\) (y°), so \(y = 103^\circ\).

1. Consecutive angles are supplementary: angle \(A\) (103°) + angle \(B\) (z°) = 180° ⇒ \(z = 180 - 103 = 77^\circ\).

1. Opposite angles are equal: angle \(B\) (z°) is equal to angle \(D\) (x°), so \(x = z = 77^\circ\).

So summarizing:

• \(x = 77^\circ\) (opposite to \(z\), and consecutive to \(103^\circ\), so supplementary? Wait, no, angle \(x\) is at vertex \(D\), which is opposite to vertex \(B\) (z°), so \(x = z\). And angle \(x\) is consecutive to angle \(y\) (103°), so \(x + y = 180\), which is \(77 + 103 = 180\), which checks out.

So:

• \(x = 77^\circ\) (opposite to \(z\), and supplementary to \(y\))

• \(y = 103^\circ\) (opposite to \(103^\circ\))

• \(z = 77^\circ\) (opposite to \(x\), and supplementary to \(103^\circ\))

Answer:

\(x = 77^\circ\), \(y = 103^\circ\), \(z = 77^\circ\)