RHS condition

The idea

If two right-angled triangles have their hypotenuses equal and one other side equal, they are congruent; this RHS (Right Hypotenuse Side) condition is a special case where SSA guarantees congruence.

A special case

We saw that two sides and an angle not between them do not always give one triangle. Now the frame has a right angle at B, with BC = 4 cm and the hypotenuse AC = 5 cm. Does this fix the triangle?

How many different triangles have these measures?

• Just one, up to a flip
• Two different ones
• Many

Find the corner A

BC = 4 cm is drawn. A is on the line through B at right angles to BC, and it must be 5 cm from C. Slide A along the line, up and down. How many places work?

Two places, one triangle

There were two places, A and A′, one above BC and one below. They are mirror images in BC, so ∆ABC and ∆A′BC fit exactly. In both, BA = 3 cm. This is different from the 30° case, where the two triangles were different.

If two right-angled triangles have their hypotenuses equal and one other side equal, they are congruent. This is the RHS (Right angle, Hypotenuse, Side) condition.

• Triangle | ∠B | BC | AC
• ABC | 90° | 4 cm | 5 cm
• A′BC | 90° | 4 cm | 5 cm

Notes

If two right-angled triangles have their hypotenuses equal and one other side equal, they are congruent. This is the RHS (Right angle, Hypotenuse, Side) condition.

Check yourself

∆ABC has ∠B = 90°, BC = 4 cm and AC = 5 cm. ∆XYZ has ∠Y = 90°, YZ = 4 cm and XZ = 5 cm. Are they congruent?

∆PQR has ∠Q = 90°, PR = 13 cm and QR = 5 cm. ∆LMN has ∠M = 90°, LN = 13 cm and MN = 5 cm. Why are they congruent?

∆ABC has ∠B = 90°, BC = 4 cm and AC = 5 cm. How many different triangles (not counting flips) can we draw?

Answer: 1

A and A′ are mirror images in BC. So the two triangles are congruent. Only one triangle (up to a flip).

Which three things does the RHS condition need?

• Yes, by RHS — correct. Yes! Both have a right angle, the hypotenuses (AC and XZ) are equal and one more side is equal.
• We cannot tell, because SSA does not always work. SSA does not always work, but with a right angle it does. This is the RHS condition.
• No, because they may be turned. Turning or flipping does not change a triangle. They are congruent.
• SAS. The right angle is not between PR and QR. SAS does not apply.
• RHS: a right angle, equal hypotenuses (PR = LN) and one equal side (QR = MN) — correct. Yes! PR and LN are opposite the right angles, so they are the hypotenuses.
• SSS, because all three sides are given. Only two sides are given. The third side comes out equal, but that is what RHS tells us.
• Two sides and the angle between them. That is SAS. RHS is for right-angled triangles.
• Three equal angles. Three equal angles only fix the shape, not the size.
• A right angle in each triangle, equal hypotenuses and one other equal side — correct. Yes! Right angle, Hypotenuse, Side.