Same angles, different sizes

The idea

Triangles with the same set of angles have the same shape but not necessarily the same size, so two triangles with equal angles need not be congruent.

Are the angles enough?

Meera and Rabia measure the three angles of the triangular frame instead of its sides: 30°, 70° and 80°. Can they now make an exact copy of the frame?

Do the three angles give just one triangle?

• Yes, one triangle only
• No, there are many triangles with these angles
• Only a very small one

Stretch the base

Here is a triangle with angles 30°, 70° and 80°. Slide the base longer and shorter. Stamp three sizes. Do the angles ever change?

Same shape, not the same size

Every size had the angles 30°, 70° and 80°. They all have the same shape. But the smaller ones cannot cover the bigger ones.

Triangles with the same set of angles have the same shape but not necessarily the same size. Two triangles that have the same angles need not be congruent.

• Base AB | Angles
• 3 cm | 30°, 70°, 80°
• 5 cm | 30°, 70°, 80°
• 8 cm | 30°, 70°, 80°

Notes

Triangles with the same set of angles have the same shape but not necessarily the same size. Two triangles that have the same angles need not be congruent.

Check yourself

Two triangles both have angles 50°, 60° and 70°. Must they be congruent?

Triangle 1 has angles 40°, 60° and 80°. Triangle 2 has angles 40° and 60° too. What is the third angle of Triangle 2, in degrees?

Answer: 80

The angles of a triangle add up to 180°. 180° − 40° − 60° = 80°. So Triangle 2 has the same three angles as Triangle 1, but it can still be a different size.

A triangular flag is copied on a photocopier at twice its size. What is true about the flag and its copy?

Rabia measures the three angles of the frame: 30°, 70° and 80°. She wants a cardboard copy of exactly the same size. What else must she know?

• Yes. Same angles means same triangle.. The angles fix only the shape. We saw triangles of base 3 cm, 5 cm and 8 cm with exactly the same angles.
• No. They may have different sizes. — correct. Yes! The same angles give the same shape, but a big triangle and a small one can have the same angles.
• Yes, if they point the same way.. Turning or flipping does not change the size. A bigger triangle will not fit on a smaller one however it is turned.
• They are congruent, because the angles are the same. Congruent figures need the same size too. The copy is twice as big.
• Their angles are different. The angles do not change when a figure is enlarged. The shape stays the same.
• They have the same angles, but they are not congruent — correct. Yes! The photocopy keeps the shape and so the angles, but its sides are twice as long, so the sizes differ.
• Nothing. The angles are enough.. The angles are not enough. A bigger or smaller triangle has the same angles.
• The length of at least one side — correct. Yes! The angles give the shape. One side length then fixes the size.
• The colour of the frame. The colour does not change the shape or the size. She needs a length.