Question

find the measures of the unknown angles in the following figures​

Answer 1

Step-by-step explanation:

Solution :-

Given that

ABCd is a Parallelogram.

BD is a diagonal

AB || CD

AD || BC

We know that

In a Parallelogram the adjacent angles are supplementary.

=> <A + <D = 180°

=> x + 45+z = 180°

=> x+z = 180°-45°

=> x+z = 135° ----------(1)

In ∆ BDC

We know that

The exterior angle is equal to the sum of the opposite interior angles

=> 60° = y+z

=> y+z = 60° -----------(2)

AB || CD ,BD is a transversal then

Alternative interior angles are equal

=> y = 45°

On Substituting the value of y in (2) then

=> 45°+z = 60°

=> z = 60°-45°

=> z = 15°

On Substituting the value of z in (1) then

=> x+15° = 135°

=> x = 135°-15°

=> x = 120°

Answer:-

The value of x = 120°

The value of y = 45°

The value of z = 15°

Used formulae:-

→ In a Parallelogram the adjacent angles are supplementary.

→ The exterior angle is equal to the sum of the opposite interior angles.

→ Alternative interior angles are equal.

Answer 2

In the given quadrilateral figure, the opposite sides are equal and diagonals are unequal. This shows that the given shape is a parallelogram ABCD where opposite angles are equal, AB || DC and AD || BC.

∴∠ADB = y = 45° [∵alternate interior angles]

Using the exterior angle property of a triangle, we have

z + y = 60°

z + 45° = 60°

z = 15°

Now, using the straight angle property,

∠BCD + 60° = 180°

∠BCD = 120°

We know that opposite angles of parallelogram are equal. Then,

x = ∠BCD = 120°

Thus, the required angles are as follows:

• Thus, the required angles are as follows:x = 120°
• Thus, the required angles are as follows:x = 120°y = 45°
• Thus, the required angles are as follows:x = 120°y = 45°z = 15°

I hope this helps you out in some way!