8.G.5: Angles, Parallel Lines and Triangles
Also written as 8.G.A.5 and CCSS.MATH.CONTENT.8.G.A.5 · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software.
The Official Wording
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. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Common Core State Standards, Grade 8, Geometry · Read it at the source · Checked October 14, 2026
What 8.G.5 Means in Plain English
Eighth graders argue angle facts informally instead of memorizing them. A transversal across parallel lines makes matching alternate angles, which is why a triangle’s three angles, like 50°, 60° and 70°, fill a straight line, and why the exterior angle beside 70° is 50° + 60° = 110°.
Students also reason that two triangles with two matching angles are similar, since the third angles must match too. These informal arguments become formal proofs in high school, about parallel lines (HSG-CO.9) and triangle angle sums (HSG-CO.10).
8.G.5 I Can Statements
- I can explain why the angles of any triangle add to 180°.
- I can find angles made when a line crosses parallel lines.
- I can use two equal angles to show two triangles are similar.
8.G.5 Worked Example
One extra line through the top vertex, drawn parallel to the base, turns the 180° fact into something students can see.
In triangle ABC, ∠A = 48° and ∠B = 67°. Line ℓ is drawn through C, parallel to AB. Use ℓ to find ∠C and explain why it works, then find the exterior angle at B.
- Side AC is a transversal across the parallel lines AB and ℓ. Alternate interior angles are equal, so the angle on ℓ to the left of C matches ∠A.angle 1 = ∠A = 48°
- Side BC is a second transversal. The angle on ℓ to the right of C matches ∠B in the same way.angle 2 = ∠B = 67°
- Angle 1, ∠C and angle 2 sit side by side along the straight line ℓ, so together they make 180°.48° + ∠C + 67° = 180°, so ∠C = 65°
- The exterior angle at B lies on a straight line with ∠B. It equals the two interior angles that do not touch it.180° − 67° = 113° = 48° + 65°
Check it: Add the interior angles: 48 + 67 + 65 = 180. Any other triangle with angles of 48° and 67° also has a 65° angle, so by angle-angle it is similar to ABC.
Common Mistakes with 8.G.5
Matching angles across lines that are not parallel
Alternate interior angles marked equal on two lines that meet off the page
Try this: Check for parallel marks or the word parallel before using any matching-angle fact. Without parallel lines, alternate angles need not be equal.
Adding the neighboring angle for an exterior angle
Exterior angle at B = ∠B + ∠C = 67° + 65° = 132°
Try this: Point to the two interior angles that do not touch the exterior angle. Only those two add to it: 48° + 65° = 113°.
Needing all three angles for similarity
“We only know two angles, so we can’t tell if the triangles are similar.”
Try this: Subtract the two known angles from 180° to show the third is forced. Two matching angles always mean three, so the triangles are similar.
Where 8.G.5 Fits
Practice 8.G.5
Quick Check
- Two angles of a triangle measure 35° and 85°. What is the third angle, and what is the exterior angle beside it?
- A transversal crosses two parallel lines, and one alternate interior angle is 72°. What is the other one, and what is the angle beside it on the same straight line?
- Triangle PQR has angles of 40° and 75°. Triangle XYZ has angles of 75° and 65°. Are they similar?