CHAPTER 5SECTION 2

Right Triangles

WARM-UPDraw and label a figure to illustrate each situation.

1) SL is an altitude of triangle RST and a perpendicular bisector of side RT.

2) Triangle HIJ is an isosceles triangle with vertex angle at J. JN is an altitude of triangle HIJ.

State whether each sentence is always, sometimes, or never true.

3) The perpendicular bisectors of a triangle intersect at more than on point.

4) The altitude of a triangle contains the midpoint of the opposite side.

5) The medians of a triangle contain the midpoints of the opposite sides.

WARM-UPDraw and label a figure to illustrate each situation.

1) SL is an altitude of triangle RST and a perpendicular bisector of side RT.

2) Triangle HIJ is an isosceles triangle with vertex angle at J. JN is an altitude of triangle HIJ.

R

S

TL

J

NIH

WARM-UPState whether each sentence is always, sometimes, or never true.

3) The perpendicular bisectors of a triangle intersect at more than on point.Never

4) The altitude of a triangle contains the midpoint of the opposite side.Sometimes

5) The medians of a triangle contain the midpoints of the opposite sides.Always

VOCABULARYLL Theorem- If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

HA Theorem- If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

VOCABULARY CONT.LA Theorem– If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent.

HL Postulate- If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

Example 1) State the additional information needed to prove each pair of triangles congruent by the given theorem or postulate.

<B and <D are right angles

A) HL

Q

M

L

V

U

T

SR

D

CB

A

B) LL

ST is congruent to TUC) LA

R

P

N

LN is congruent to QR or NM is congruent to RP

Example 2) Find the values of x and y so that triangle MNO is congruent to triangle RST.

15 cm

58

RS

T

(47 – 8x) cm

M

ON

(3y – 20)Assume triangle MNO is congruent to triangle RST. Then <N is congruent to <R and MO is congruent to RT.

m<N = m<R58 = 3y – 2078 = 3y26 = y

MO = RT47 – 8x = 15-8x = -32x = 4

Example 3) Find the values of x and y so that triangle JKL is congruent to triangle MLK.

L

(2x - 15)

J

M

K

(3 – 10x)

Assume triangle JKL is congruent to triangle MLK.

m<JLK = m<LKM8y = 10y – 8-2y = -8y = 4

JK = LM3 – 10x = 2x - 153 - 12x = -15-12x = -18x = 3/2 or 1.5

(8y)(10y - 8)

Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate.

A

X

Z Y

B C

A) AB = 2x + 6, BC = 15, AC = 3x + 4, XY = 20, YZ = x + 8; LL

Let XY be congruent to AB and YZ be congruent to BC.

XY = AB20 = 2x + 614 = 2x7 = x

YZ = BCx + 8 = 15x = 7

So x is equal to 7.

Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate.

A

X

Z Y

B C

B) m<Z = 55, BC = 15x + 2, m<C = 55, AB = 24, YZ = 4x + 13; LA

Let <Z be congruent to <C and YZ be congruent to BC.

YZ = BC4x + 13 = 15x + 213 = 11x + 211 = 11x1 = x

So x is equal to 1.

Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate.

A

X

Z Y

B C

C) YX = 21, m<X = 9x + 9, AB = 21, m<A = 11x - 3; LA

Let XY be congruent to AB and <X be congruent to <A.

m<X = m<A9x + 9 = 11x - 39 = 2x – 312 = 2x6 = x

So x is equal to 6.

Example 4) Find the value of x so that triangle ABC is congruent to triangle XYZ by the indicated theorem or postulate.

A

X

Z Y

B C

D) AC = 28, AB = 7x + 4, ZX = 9x + 1, YX = 5(x + 2); HL

Let AC be congruent to ZX and AB be congruent to YX.

XY = AB5(x + 2) = 7x + 45x + 10 = 7x + 410 = 2x + 46 = 2x3 = x

AC = ZX28 = 9x + 127 = 9x3 = x

So x is equal to 3.