Sec 4'4 Prove Triangles Congruent by SAS and HL

1
Sec 4.4 Prove Triangles Congruent by SAS and HL

• The angle that is formed by two sides is called
  the included angle

Postulate 20 Side-Angle-Side (SAS) Congruence
Postulate

• If two sides and the included angle (the angle
  between) of one triangle are congruent to two
  sides and the included angle of a second
  triangle, then the two triangles are congruent.

If Side AC ? XZ, Angle ?C ?
?Z, Side CB ? ZY Then
?ACB ? ?XZY
2
EXAMPLE 1
Use the SAS Congruence Postulate
Write a proof.
GIVEN
PROVE
3
EXAMPLE 2
Use SAS and properties of shapes
In the diagram, QS and RP pass through the center
M of the circle. What can you conclude about
MRS and MPQ?
SOLUTION
4

• The two sides of a triangle that form an angle
  are adjacent to the angle.
• The side not adjacent to the angle is opposite
  the angle.
• If you know the lengths of two sides and the
  measure of an angle that is not included between
  them, you can create two different triangles.

• Therefore, SSA (I know what youre thinking!) is
  NOT a valid method for proving that triangles are
  congruent, but there is a special case for right
  triangles.

5

• In a right triangle, the sides adjacent to the
  right angle are called the legs.
• The side opposite the right angle is called the
  hypotenuse of the right triangle.

• Theorem 4.5 Hypotenuse-Leg (HL) Congruence
  Theorem
• If the hypotenuse and a leg of a right triangle
  are congruent to the hypotenuse and a leg of a
  second right triangle, then the two triangles are
  congruent.

6
Write a proof.
EXAMPLE 3
Redraw the triangles so they are side by side
with corresponding parts in the same position.
Mark the given information in the diagram.
7
EXAMPLE 3
8
Bookworkp.243-2453-16 all, 18-22even,
25-27all, 34, 36