Problem Solving with Interior and Exterior Angles
Problem Solving with Interior and Exterior Angles is a Grade 7 math skill in Big Ideas Math Advanced 2, Chapter 3: Angles and Triangles, where students combine the triangle angle-sum theorem (interior angles sum to 180 degrees) with the Exterior Angle Theorem to find unknown angle measures in diagrams involving triangles with marked interior and exterior angles. This integrates two theorems into a multi-step problem-solving framework.
Key Concepts
Property When angles are given as algebraic expressions, use the Triangle Angle Sum Theorem ($m\angle A + m\angle B + m\angle C = 180^\circ$) or the Exterior Angle Theorem to set up an equation and solve for the variable $x$.
Examples The angles of a triangle measure $x^\circ$, $(x+20)^\circ$, and $(x+40)^\circ$. Set up the equation: $x + (x+20) + (x+40) = 180$. Simplifying gives $3x + 60 = 180$, so $x = 40$. The angles are 40°, 60°, and 80°. In a right triangle, the acute angles are $x^\circ$ and $(2x)^\circ$. Since they are complementary: $x + 2x = 90$, leading to $3x = 90$ and $x = 30$. The angles are 30° and 60°. An exterior angle measures $(4x + 10)^\circ$. The remote interior angles measure $(x + 20)^\circ$ and $(2x + 10)^\circ$. Equation: $(x + 20) + (2x + 10) = 4x + 10$. Solving gives $3x + 30 = 4x + 10$, so $x = 20$.
Explanation In Integrated Math, geometry and algebra are heavily combined.
Common Questions
How do you find an unknown interior angle of a triangle?
Use the triangle angle-sum theorem: all three interior angles add up to 180 degrees. If two angles are known, subtract their sum from 180 to find the third.
How do you find an unknown exterior angle?
The exterior angle equals the sum of the two non-adjacent remote interior angles (Exterior Angle Theorem). If both remote interior angles are known, add them.
How do you combine both theorems in a single problem?
Identify which angle you can find first using one theorem, then use that result as a known value in the other theorem. Multi-step problems often require applying both theorems in sequence.
What is Big Ideas Math Advanced 2 Chapter 3 about?
Chapter 3 covers Angles and Triangles, including the triangle angle-sum theorem, the Exterior Angle Theorem, problem solving with angle relationships, and classifying triangles by angle type.