In a triangle HJK, HJ = HK. G is a point on HJ such that HG = GK = JK. What is the degree measure of two-third of (∠HGK + ∠GKJ)?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. 96°.
The sum of the interior angles of a triangle is 180°.
ΔHJK is an isosceles triangle where HJ = HK
Also, HG = GK = JK
Let, ∠KHG = ∠GKH = y
So, ∠KGJ = 2y [Exterior angle of ΔHGK]
Now, GK = JK
So, ∠KGJ = ∠GJK = 2y
Also, HJ = HK then, ∠GJK = ∠HJK = ∠JKH = 2y
In ΔHJK, ∠KHG + ∠HJK + ∠JKH = 180°
⇒ y + 2y + 2y = 180°
⇒ 5y = 180°
⇒ y = 180°/5 = 36°
∠HGK = 180° - ∠KGJ = 180° - 2 × 36° = 180° - 72° = 108°
∠GKJ = ∠JKH - ∠GKH = 2y - y = y = 36°
Now, (2/3) of (∠HGK + ∠GKJ) = (2/3) × (108° + 36°) = (2/3) × 144° = 96°
Therefore, the required value is 96°.