Triangle Angle Sum and Exterior Angles

Students prove that the interior angles of any triangle sum to 180° using parallel line properties, and apply the exterior angle theorem (an exterior angle equals the sum of the two non-adjacent interior angles) to find missing angle measures.

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Grade 8 · Geometry: Transformations · question 1 of 3

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The three interior angles of a triangle sum to 180 degrees.

The largest angle is the one with the biggest share.

Read it: A triangle's angles and its exterior angle

Triangle PQR has P = 57° and R = 64°. Extend side PQ past Q to D: the angle outside Q is 57 + 64 = 121°. Q itself is 180 − 121 = 59°.

The three angles of a triangle add to 180°.
The angle outside a corner and the angle at that corner make a straight line, 180°.
So the exterior angle equals the two far interior angles added.

Both 180s share the angle at Q, so what is left with it must match: the outside angle equals P + R.

You can find the exterior angle without finding Q first.

An exterior angle is always larger than either far angle.

An exterior angle of a triangle equals the sum of the two far interior angles.

Taking the exterior angle as 180 − one far angle: 180 − 57 = 123.

The exterior angle sits beside Q, not P. It is 180 − Q, which is P + R: 57 + 64 = 121.

A triangle has angles of 67° and 29°. One side is extended past the third corner. What is the exterior angle there, in degrees? — 96. An exterior angle equals the two far interior angles added: 67 + 29 = 96.

Triangle with two angles 50° and 60° — find the third interior angle and an exterior angle

Today we'll explore two big ideas about triangle angles. Here's our triangle, with corners labeled A, B, and C.

The angle at A measures fifty degrees, and the angle at B measures sixty degrees.

But what about the angle at corner C? How big is it?

Grade 8 · Interior angles sum to 180°, exterior = sum of the two remote interior angles

Here's our first key idea: the three interior angles of any triangle always add up to one hundred eighty degrees.

So we just subtract the two we know from one hundred eighty.

The angle at C is seventy degrees.

Now watch what happens when we extend side BC straight past corner C.

The new angle that opens up outside the triangle is called an exterior angle.

The interior seventy and the exterior angle sit on a straight line, so together they must equal one hundred eighty degrees.

One hundred eighty minus seventy gives us our exterior angle.

The exterior angle is one hundred ten degrees.

Look closely — one hundred ten is exactly fifty plus sixty, the two angles farthest from it.

That's the Exterior Angle Theorem: an exterior angle equals the sum of the two non-adjacent interior angles.

Missing interior angle: seventy degrees. Exterior angle: one hundred ten degrees. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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Standard

CCSS.8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles

Grade 8

Quoted from the Common Core State Standards.

Where this sits

Comes after

Angle Relationships with Parallel Lines and Transversals

Leads to

Angle-Angle Criterion for Triangle Similarity

What a star here means

A star here is the top of a twenty-rung ladder, in four blocks of equal length — three of facts and one of the skill used in a real question — where a right answer climbs one rung and a wrong one drops back to the start of the block, never below it. The star is kept for good. How we grade

All of Grade 8

This is one rung of 48 in Grade 8 mathematics, in the order they build on each other.

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