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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
When a transversal crosses two parallel lines it creates eight angles, but only two different measures. Corresponding angles are equal, alternate interior angles are equal, and angles that sit side by side on a straight line add to 180°. Eighth graders establish these facts with informal arguments, often by translating one intersection onto the other along the transversal, and then use them as tools.
The payoff is a set of triangle facts. Tear the three corners off a paper triangle and line them up: they form a straight angle, so the interior angles sum to 180°. A tidier argument draws a line through one vertex parallel to the opposite side and uses alternate interior angles. From there the exterior angle of a triangle equals the sum of the two remote interior angles. Finally, if two angles of one triangle match two angles of another, the third angles must match too, so the triangles are similar. This angle-angle criterion becomes the quickest test for similarity.
Corresponding angles are only equal when the lines are parallel. Students should look for the parallel markings or the given information before using the rule.
An exterior angle equals the sum of the two remote interior angles, not including the angle right next to it.
Students sometimes check all three angles. Two matching angles are enough, because the third is then forced by the 180° sum.
Same-side interior angles add to 180° rather than being equal. Mark the angles on the diagram to keep the relationships straight.
In triangle PQR, angle P = 48° and angle Q = 67°. Side QR is extended past R to form an exterior angle at R. Find angle R and the exterior angle.
Answer: Angle R = 65° and the exterior angle at R = 115°.
The torn-corner activity for the angle sum and a tracing-paper translation along a transversal for corresponding angles give students physical evidence before any argument is written. Then ask them to turn the evidence into a few sentences that name the relationships used, which is the informal argument the standard asks for.
Assessments mix numerical angle-finding with explanation items, such as choosing the reason that justifies a step. For angle-angle similarity, a good task gives two triangles with two labelled angles each and asks whether they must be similar, and why.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 70 (also accepted: 70°)
The angles sum to 180°, so the third angle is 180 - 72 - 38 = 70°.
Answer: A) 115°
Corresponding angles are equal when the lines are parallel, so it is also 115°.
Answer: 75 (also accepted: 75°)
The exterior angle equals the sum of the two remote interior angles: 130 - 55 = 75°.
Answer: C) Yes, by angle-angle
Triangle A's third angle is 180 - 40 - 60 = 80°. So both triangles have angles 40°, 60° and 80°, and two matching angles make them similar.
Answer: 45 (also accepted: 45°)
Same-side interior angles add to 180°: x + 3x = 180, so 4x = 180 and x = 45°.
Answer: A) Draw a line through one vertex parallel to the opposite side; the alternate interior angles and the angle at the vertex form a straight line
The parallel line creates alternate interior angles equal to the triangle's base angles. Together with the top angle they fill a straight line, which is 180°.
A full lesson with slides, activities and an exit ticket on angles in triangles and parallel lines, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.A.5 with an answer key, ready in about a minute.
Make a worksheet →Turn angles in triangles and parallel lines into a quiz students answer online that marks itself, with a class summary for you.
Build a test →If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. The third angles must also match because each triangle's angles sum to 180°.
No. The standard asks for informal arguments, clear explanations that cite angle facts. Formal proofs come in high school geometry.