# PROBLEM LINK

**Author:** Mann Mehta

**Tester:** Jaymeet Mehta

**Editorialist:** Vatsal Parmar

# DIFFICULTY

Easy

# PREREQUISITES

Basic Math, Triangle Properties, Geometry

# PROBLEM

You are given a **9-Queries** on an Isosceles Triangle, in which you get to know isosceles sides and any one angle of an Isosceles Triangle. Your task is to predict other two angles of Isosceles Triangle.

# EXPLANATION

Here there are total 3-ways such that we select 2 side as an isosceles side in a triangle. That's why total 9-queries are possible for each angel with all 3 ways.*(3-Angles × 3-Ways = 9-Queries)*

**What is Isosceles Triangle ?**

In geometry, an isosceles triangle is a triangle that has **at least two sides** of equal length.

**angle ( A ) + angle ( B ) + angle ( C ) = 180If AB = AC then angle(B) = angle(C)**

Now, we go through first three queries Where $AB$ and $AC$ are isosceles side therefore **angle(B) = angle(C)** .

**1) Query-1**

Let’s consider angle(A) which is given as A and suppose other two angles angle(B) and angle(C) as X.

Now applying the angle sum property.

**angle(A)+angle(B)+angle(C)=180**

** A + X + X = 180 **

** 2X = 180 − A**

** X = (180 − A) / 2**

*2) Query-2*Let's consider

**angle ( B )**which is given as

**B**and suppose other two angles

**angle(A)**as

**X**and

**angle(C)**as

**Y**.

Now applying the angle sum property.

**angle(A)+angle(B)+angle(C)=180**

** X + B + Y = 180 **

We know that **angle(B) = angle(C)** therefore **angle(C)=B**

** X + B + B = 180**

** X + 2B = 180**

** X = 180 − 2B**

**Now in this query angle ( B ) and angle ( C ) is an isosceles angle’s therefore to make a valid triangle it’s value must be in range :- 0 < angle ( B ), angle ( C ) < 90 **

**3) Query-3**

Let’s consider angle ( C ) which is given as C and suppose other two angles angle ( A ) as X and angle(B) as Y.

Now applying the angle sum property.

**angle(A)+angle(B)+angle(C)=180**

** X + Y + C + = 180 **

We know that **angle(B) = angle(C)** therefore **angle(B)=C**

** X + C + C + = 180**

** X + 2C = 180**

** X = 180 − 2C**

**Now in this query angle ( B ) and angle ( C ) is an isosceles angle’s therefore to make a valid triangle it’s value must be in range :- 0 < angle ( B ),angle ( C ) < 90 **

**Now similarly go for other queries also.**

# TIME COMPLEXITY

Time Complexity is**O(1)**per test case.