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Matching the Corners

Part of the Triangles section of Coddy's Geometry journey. Lesson 44 of 60.

You do not need a statement like △ABC ~ △DEF to know two triangles are similar. Two pairs of equal angles are enough. If two angles match, the third angles match as well, since each is 180 minus the other two. This is AA similarity.

$A small triangle ABC and a bigger turned triangle PQR, with matching arcs at A and P and at B and Q$

Then match the corners by the marks, not by how the triangles are drawn. Here one arc sits at A and at P, two arcs at B and at Q. So A matches P, B matches Q, C matches R, and △ABC ~ △PQR even though PQR is turned over. Side AB matches PQ.

One pair of equal angles is not enough: the other two angles could share what is left in many ways.

Challenge

The arcs mark two pairs of equal angles in △ABC and △PQR. Is there enough to find PQ? Answer, and if it is, find PQ. Press Check after each.

Try it yourself

$Is there enough information?$

Is there enough information?

All lessons in Triangles

1The Triangle Inequality

Can Three Sticks Meet?Exactly Reaching The Range of the Third Side Recap: The Third Side

4Area of a Triangle

Half the Rectangle Height Is Not a Side Right Triangle Area Finding the Height Any Side Can Be the Base Recap: Area

7Is It Right?

The Converse Testing a Triangle Acute or Obtuse?Sorting by the Test Recap: Right or Not

10Scale in the World

Shadows Across the River Nested Triangles A Side with an Expression Recap: Scale in the World

2Sides and Names

Sorting by Sides The Longest Side Ticks into Equations Recap: Sorting Triangles

5The Pythagorean Theorem

Squares on the Sides Finding the Hypotenuse Pythagorean Triples When It Is Not Whole Recap: The Hypotenuse

8Special Right Triangles

The 45-45-90 Triangle Legs from the Hypotenuse The 30-60-90 Triangle All Three Sides Which Triangle Is It?Recap: Special Triangles

11Two Tools at Once

Height of an Isosceles Area from the Height The Equilateral Triangle Right Triangle Perimeter Recap: Two Tools

3Perimeter

Around the Edge A Missing Side Isosceles Perimeter Sides as Expressions Recap: Around the Edge

6Finding a Leg

Working Backwards The Ladder Which Side Is c?A Leg That Is Not Whole Recap: Finding a Leg

9Similar Triangles

Same Shape, Different Size The Scale Factor Finding a SideMatching the CornersRecap: Similar Triangles

12Final Challenges

Final: Three Sticks Final: The Fence Final: The Ladder Final: The Shadow Final: Master of Triangles