Quadrilaterals · Lesson 4 of 13
Angles in a Quadrilateral
“Derive and apply the 360° interior-angle sum of quadrilaterals.”
• Explain the 360° interior-angle sum. • Find missing quadrilateral angles. • Use angle sums to test a proposed figure. • Explain why three right angles force a rectangle.
Try drawing a four-sided figure with three right angles and a fourth angle that is different. You can move the last side around, but closing the figure while keeping those first three corners is the difficulty. The corners are connected by a total that cannot change. To discover that total, you do not need a new measuring tool. You can use something you already know: the three angles of a triangle add to 180°.
A diagonal will help us break one unfamiliar problem into two familiar ones. Watch what happens to the corner angles when the quadrilateral becomes two triangles.
4.2 Angles in a Quadrilateral
Take convex quadrilateral SOME, with its letters in boundary order. Draw diagonal SM. The two triangles are SOM and SEM. The angles at S and M are split into two parts, while the angles at O and E stay whole.
Add the triangle angle sums. The pieces at S recombine to the original angle at S, and the pieces at M recombine to the original angle at M. No angle is lost or counted twice.
The argument does not require equal sides, parallel sides or right angles. It applies to every simple quadrilateral. For an inward-pointing, concave quadrilateral, use the diagonal that lies inside the figure and count the reflex interior angle correctly; we will revisit that case in the mixed lesson.
When three angles are known, first add those three, then subtract their total from 360°. Include the degree sign and check by adding all four angles again.
Problem
Three angles are 75°, 95° and 110°. Find the fourth.
Problem
The angles are x, x, 80° and 100°. Find x.
Problem
Three corners of a quadrilateral are right angles. What is the fourth?
Problem
Could a quadrilateral have four angles of 100°?
Quiz
The sum of interior angles of any quadrilateral is:
Three angles are 90°, 80° and 100°. The fourth is:
Can a quadrilateral have angles 100°, 100°, 100°, 100°?
Why does drawing a diagonal help prove the quadrilateral angle sum?
A quadrilateral has three right angles. The fourth must be:
Practice Problems
- Find the fourth angle if the other three are 65°, 115° and 90°.
- Three equal angles are each 85°. Find the fourth.
- Angles are x, x, 80° and 100°. Find x.
- Would 50°, 80°, 100° and 140° satisfy the quadrilateral angle-sum rule?
- Explain why a quadrilateral with all four angles equal must be a rectangle.
1. Known total = 65° + 115° + 90° = 270°. 2. Fourth angle = 360° − 270° = 90°.
1. Three angles total 3 × 85° = 255°. 2. The fourth is 360° − 255° = 105°.
1. 2x + 180° = 360°, so 2x = 180°. 2. Dividing by 2 gives x = 90°.
1. Their sum is 370°. 2. This differs from 360°, so they cannot be the four interior angles.
1. Let each angle be x. Then 4x = 360°. 2. So x = 90°. 3. Four right angles satisfy the rectangle definition; this does not alone force four equal sides.
Key Takeaways
• A suitable diagonal splits a quadrilateral into two triangles. • Its four interior angles therefore total 360°. • Find a missing angle by subtracting the known total from 360°. • Three right angles force the fourth to be a right angle. • Equal angles imply a rectangle, but do not alone imply a square.