Find the remaining angles in the following trapeziums—

Question

Homi · Accepted Answer

We will use the properties of parallel lines and trapeziums to find the unknown angles.

Step 1 — Finding angles in the first trapezium

Let us look at the first trapezium. The arrows on the top and bottom sides tell us these sides are parallel. When two parallel lines are cut by another line (called a transversal), the angles between the parallel lines on the same side of the transversal are called consecutive interior angles. These angles always add up to $\mathbf{180^\circ}$ .

Let the bottom-left angle be $D$ and the bottom-right angle be $C$ . Let the top-left angle be $A$ and the top-right angle be $B$ . From the diagram, we are given: Angle $D = 135^\circ$ Angle $C = 105^\circ$

Since the top and bottom sides are parallel, angle $A$ and angle $D$ are consecutive interior angles. So, their sum is $180^\circ$ .

$A + D = 180^\circ$

$A + 135^\circ = 180^\circ$

$A = 180^\circ - 135^\circ$

$\boxed{A = 45^\circ}$

Similarly, angle $B$ and angle $C$ are consecutive interior angles. So, their sum is $180^\circ$ .

$B + C = 180^\circ$

$B + 105^\circ = 180^\circ$

$B = 180^\circ - 105^\circ$

$\boxed{B = 75^\circ}$

Diagram 1

Step 2 — Finding angles in the second trapezium

Let us look at the second trapezium. The arrows on the top and bottom sides tell us these sides are parallel. The tick marks on the left and right sides mean these two non-parallel sides are equal in length. This type of trapezium is called an isosceles trapezium. In an isosceles trapezium, the base angles (angles on the same parallel side) are equal.

Let the bottom-left angle be $D$ and the bottom-right angle be $C$ . Let the top-left angle be $A$ and the top-right angle be $B$ . The diagram shows one of the bottom angles is $\mathbf{100^\circ}$ . Let us assume this is angle $D$ .

Since it is an isosceles trapezium, the base angles $C$ and $D$ are equal.

$C = D$

$C = 100^\circ$

$\boxed{C = 100^\circ}$

Now, we use the property of consecutive interior angles, just like in Step 1. Angle $A$ and angle $D$ are consecutive interior angles, so their sum is $180^\circ$ .

$A + D = 180^\circ$

$A + 100^\circ = 180^\circ$

$A = 180^\circ - 100^\circ$

$\boxed{A = 80^\circ}$

Similarly, angle $B$ and angle $C$ are consecutive interior angles, so their sum is $180^\circ$ .

$B + C = 180^\circ$

$B + 100^\circ = 180^\circ$

$B = 180^\circ - 100^\circ$

$\boxed{B = 80^\circ}$

Diagram 2

Answer

(i) The remaining angles are $\mathbf{45^\circ}$ and $\mathbf{75^\circ}$ . (ii) The remaining angles are $\mathbf{80^\circ}$ , $\mathbf{80^\circ}$ , and $\mathbf{100^\circ}$ .

Question 11

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