E. Spatial Sense · Spatial Sense: Shapes, Grids, Angles, Area and Surface AreaInterior and Exterior Angles of Polygons
E2.3 A1About 45 minutes
Learning goals- I can use the fact that the interior angles of a triangle add to 180° and of a quadrilateral add to 360°.
- I can find an exterior angle from the interior angle next to it.
- I can solve for unknown angles in triangles and quadrilaterals and explain my reasoning.
Words to know
interior angle
An angle inside a polygon, between two sides that meet at a vertex.
exterior angle
The angle outside a polygon between one side and the next side extended in a straight line. An interior angle and its exterior angle are supplementary (they add to 180°).
polygon
A closed flat shape with straight sides, like a triangle, quadrilateral or pentagon.
regular polygon
A polygon with all sides equal and all angles equal.
Remember from Grade 5
In Grade 5 you found a missing angle in a triangle by subtracting from 180°, like 180° − 35° − 45° = 100°. You also learned that an isosceles triangle has two equal sides, and its angles opposite those sides are equal too. Today you'll look at why 180° works, and go further: quadrilaterals and exterior angles.
Let's start with a paper triangle
Cut out any triangle (a grown-up can help with scissors). Tear off its three corners and line them up side by side, points touching. They always make a straight line: 180°! However you draw the triangle, the three interior angles add to 180°.
Now extend one side of the triangle past a vertex, like a road going straight on. The angle between the extended side and the next side is an exterior angle. The interior angle and the exterior angle sit together on a straight line, so they are supplementary: they add to 180°.
A shortcut for triangles
Look at the picture again. The exterior angle at B (110°) equals the sum of the two interior angles at the other vertices (50° + 60°). This always works for triangles: both 110° and 50° + 60° are what's left when you take 70° from 180°.
Quadrilaterals add to 360°
Draw a diagonal in any quadrilateral. It splits the shape into 2 triangles. The angles of the triangles fill the corners of the quadrilateral exactly, so the interior angles of a quadrilateral add to 2 × 180° = 360°.
The same idea works for bigger polygons: a pentagon splits into 3 triangles, so its angles add to 3 × 180° = 540°.
Worked exampleExample 1: A missing angle in a triangle
A triangle has angles of 48° and 67°. Find the third angle.
- The interior angles add to 180°.
- 48° + 67° = 115°.
- 180° − 115° = 65°.
Worked exampleExample 2: A missing angle in a quadrilateral
A quadrilateral has angles of 90°, 85° and 110°. Find the fourth angle.
- The interior angles add to 360°.
- 90° + 85° + 110° = 285°.
- 360° − 285° = 75°.
Worked exampleExample 3: Exterior angles
Triangle PQR has interior angles P = 45° and Q = 80°. Find the interior angle at R and the exterior angle at R.
- Interior angle at R: 180° − 45° − 80° = 55°.
- Exterior angle at R: supplementary to the interior angle, so 180° − 55° = 125°.
- Check with the shortcut: 45° + 80° = 125° ✓.
Interior 55°, exterior 125°.
Walking around a shape
Imagine walking around a triangle on a big field. At each corner you turn by the exterior angle. When you get back to the start, you are facing the way you began: you've made one full turn. So the exterior angles of any polygon (one at each vertex) add to 360°. (This is true for polygons with no dents, called convex polygons, like all the shapes in this lesson.)
For the triangle in the picture: 130° + 110° + 120° = 360° ✓ (180° − 50°, 180° − 70° and 180° − 60°).
Just for funWalk all the way around a triangle, a rectangle or even a stop-sign shape, and your turns always add up to 360°. You did a full spin without ever getting dizzy!
Write your reasons
When you find an angle, write the reason beside it: "angles in a triangle add to 180°", "supplementary", "opposite angles". Your reasons let someone else follow your thinking, and they help you catch mistakes. This is how mathematicians communicate.
Hands-on activityTear and turn
You need: paper, a ruler, a pencil and scissors (with a grown-up's help), and some open floor space.
- Draw a large triangle with a ruler and cut it out. Mark each corner with a dot.
- Tear off the three corners. Put the dots together so the corners sit side by side. Do they make a straight line?
- Do the same with a quadrilateral. What do the four corners make? (A full turn, all the way around the point.)
- Use painter's tape or string to make a big triangle on the floor (with a grown-up's OK). Walk around it. At each corner, notice how much you turn. When you get back to the start, how much have you turned altogether?
Math talk: "Why do the corners of a triangle always make a straight line, no matter what triangle you draw?"
Your turn
1A triangle has angles of 35° and 85°. Find the third angle.
60°. 180° − 35° − 85° = 60°.
2A quadrilateral has angles of 100°, 80° and 95°. Find the fourth angle.
85°. 360° − 100° − 80° − 95° = 85°.
3An interior angle of a polygon is 125°. What is the exterior angle at that vertex?
4A triangle has interior angles of 40° and 75°. What is the exterior angle at the third vertex?
115°. The third interior angle is 180° − 40° − 75° = 65°, and 180° − 65° = 115°. (Shortcut: 40° + 75° = 115°.)
5What is each exterior angle of a rectangle? What do they add to?
Each is 90° (180° − 90°). They add to 4 × 90° = 360°.
6What is each interior angle of an equilateral triangle? Each exterior angle?
Interior: 180° ÷ 3 = 60°. Exterior: 180° − 60° = 120°.
More practice: warm-up
1The interior angles of a triangle add to how many degrees?
2The interior angles of a quadrilateral add to how many degrees?
3An interior angle and its exterior angle add to how many degrees?
4What is each interior angle of a square?
5The exterior angles of any polygon (one at each vertex) add to how many degrees?
More practice: on your own
1A right triangle has one angle of 32°. Find the other two angles.
90° and 58°. One angle is the right angle (90°), and 180° − 90° − 32° = 58°.
2An isosceles triangle has two equal angles of 72°. Find the third angle.
36°. 180° − 72° − 72° = 36°.
3One angle of a parallelogram is 65°. Find the other three.
115°, 65° and 115°. The opposite angle is equal (65°). The other two are equal to each other and the four angles add to 360°: 360° − 65° − 65° = 230°, and 230° ÷ 2 = 115°.
4A kite has two equal angles of 100° and one angle of 90°. Find the fourth angle.
70°. 360° − 100° − 100° − 90° = 70°.
5Two exterior angles of a triangle are 110° and 130°. Find the third exterior angle and all three interior angles.
Third exterior angle: 360° − 110° − 130° = 120°. Interior angles: 180° − 110° = 70°, 180° − 130° = 50° and 180° − 120° = 60°. Check: 70° + 50° + 60° = 180° ✓.
Challenge
1What do the interior angles of a pentagon add to? How big is each angle of a regular pentagon?
Diagonals drawn from one vertex split a pentagon into 3 triangles, so the angles add to 3 × 180° = 540°. A regular pentagon has 5 equal angles: 540° ÷ 5 = 108° each.
2Use exterior angles to find each interior angle of a regular hexagon.
The 6 exterior angles add to 360°, so each is 360° ÷ 6 = 60°. Each interior angle is 180° − 60° = 120°.
3In a triangle, the exterior angle at C is 140° and the interior angle at A is 65°. Find the interior angles at B and C.
Interior angle at C: 180° − 140° = 40°. Interior angle at B: 180° − 65° − 40° = 75°. (Shortcut check: 65° + 75° = 140° ✓, the exterior angle.)
Quiz
1. A triangle has angles of 90° and 25°. What is the third angle?
2. A quadrilateral has angles of 120°, 60° and 90°. What is the fourth angle?
3. An interior angle is 72°. What is the exterior angle at the same vertex?
4. Why do the angles of a quadrilateral add to 360°?
5. Two interior angles of a triangle are 50° and 60°. What is the exterior angle at the third vertex?
6. What do the exterior angles of a triangle (one at each vertex) add to?
Big idea: E2.3 includes using the properties of interior and exterior angles to solve for unknown angles. Key facts: triangle angles add to 180°, quadrilateral angles add to 360°, an interior angle and its exterior angle add to 180°, and the exterior angles of any polygon add to 360°.
Common difficulties:
- Using 180° for quadrilaterals. Have your child draw a diagonal to see the two triangles.
- Confusing exterior angles with reflex angles. An exterior angle is between one side and the extension of the next side, not the angle all the way around the outside.
- Skipping reasons. Ask "How do you know?" for each step.
Social-emotional learning (A1): The tear-and-turn activity lets children discover the rule themselves, which builds confidence. Praise careful explanations.
How to help: Do the tearing activity together with different triangles (supervise scissors). Ask your child to teach you why it works.