Question

Question 13: Solve for x here. It is a rectangle. 100 points to whoever gets it.

Answer 1

x = 12

Explanation:

we are here given a rectangle which has an unknown angle (5x+7)° . We need to solve out for x . Firstly we know that the diagonals of a rectangle are equal and they bisect each other.

Also we know that in a triangle, angles opposite to equal sides are also equal. Hence here , we can say that ,

mHGD = mHDG = (5x+7)° .

again we know that the measure of a straight line is 180° . So we can find out the measure of angle DHG as ,

mDHG = 180° - 134°

mDHG = 46°

again we know that the angle sum property of a triangle is 180° . Therefore in ∆DHG , we have;

5x + 7 + 5x + 7 + 46 = 180

10x + 60 = 180

10x = 120

x = 12

and we are done!

Answer 2

The value of x is 12.

Explanation:

Angles around a point sum to 360°.

According to the Vertical Angle Theorem, when two straight lines intersect, the opposite vertical angles are congruent.

Therefore:

• m∠DHE = m∠GHF = 134°
• m∠DHG = m∠EHF = 180 - 134° = 46°

The diagonals of a rectangle are equal in length and bisect each other.

Therefore:

• DH = HF = GH = HE

This means that triangle DHG is an isosceles triangle with base DG.

Therefore:

• m∠HDG = m∠HGD = (5x + 7)°

Interior angles of a triangle sum to 180°. Therefore:

⇒ m∠HDG + m∠HGD + m∠DHG = 180°

⇒ (5x + 7)° + (5x + 7)° + 46° = 180°

⇒ 5x + 7 + 5x + 7 + 46 = 180

⇒ 10x + 60 = 180

⇒ 10x = 120

⇒ x = 12

Therefore, the value of x is 12.