Transcript Angle Relationships in Triangles

Angle Relationships in Triangles
First 10!
Step 1:
Draw a triangle and label the vertices ABC.
Step 2:
Measure each angle with a protractor.
Step 3:
Calculate the sum of the angle measures.
Step 4:
Did your neighbor find the same thing?
Holt McDougal Geometry
Angle Relationships in Triangles
Holt McDougal Geometry
Angle Relationships in Triangles
A corollary is a theorem whose proof follows
directly from another theorem. Here are two
corollaries to the Triangle Sum Theorem.
Holt McDougal Geometry
Angle Relationships in Triangles
Example 1: Application
After an accident, the positions
of cars are measured by law
enforcement to investigate the
collision. Use the diagram
drawn from the information
collected to find mXYZ.
mXYZ + mYZX + mZXY = 180°
mXYZ + 40 + 62 = 180
mXYZ + 102 = 180
mXYZ = 78°
Holt McDougal Geometry
Sum. Thm
Substitute 40 for mYZX and
62 for mZXY.
Simplify.
Subtract 102 from both sides.
Angle Relationships in Triangles
Example 1: Application
After an accident, the positions
of cars are measured by law
enforcement to investigate the
collision. Use the diagram
drawn from the information
collected to find mYWZ.
118°
Step 1 Find mWXY.
mYXZ + mWXY = 180°
62 + mWXY = 180
mWXY = 118°
Holt McDougal Geometry
Lin. Pair Thm. and  Add. Post.
Substitute 62 for mYXZ.
Subtract 62 from both sides.
Angle Relationships in Triangles
Example 1: Application Continued
After an accident, the positions of
cars are measured by law
enforcement to investigate the
collision. Use the diagram drawn
from the information collected
to find mYWZ.
118°
Step 2 Find mYWZ.
mYWX + mWXY + mXYW = 180°
Sum. Thm
mYWX + 118 + 12 = 180 Substitute 118 for mWXY and
12 for mXYW.
mYWX + 130 = 180 Simplify.
mYWX = 50° Subtract 130 from both sides.
Holt McDougal Geometry
Angle Relationships in Triangles
Example 2
The measure of one of the acute angles in a
right triangle is 63.7°. What is the measure of
the other acute angle?
Let the acute angles be A and B, with mA = 63.7°.
mA + mB = 90°
Acute s of rt.
63.7 + mB = 90
Substitute 63.7 for mA.
mB = 26.3°
Holt McDougal Geometry
are comp.
Subtract 63.7 from both sides.
Angle Relationships in Triangles
The interior is the set of all points inside the
figure. The exterior is the set of all points
outside the figure. It is formed by one side of the
triangle and extension of an adjacent side.
Exterior
Interior
Holt McDougal Geometry
Angle Relationships in Triangles
Each exterior angle has two remote interior
angles. A remote interior angle is an interior
angle that is not adjacent to the exterior angle.
4 is an exterior angle.
Exterior
Interior
The remote interior
angles of 4 are 1
and 2.
3 is an interior angle.
Holt McDougal Geometry
Angle Relationships in Triangles
Example 3: Applying the Exterior Angle Theorem
Find mB.
mA + mB = mBCD
Ext.  Thm.
15 + 2x + 3 = 5x – 60
Substitute 15 for mA, 2x + 3 for
mB, and 5x – 60 for mBCD.
2x + 18 = 5x – 60
78 = 3x
Simplify.
Subtract 2x and add 60 to
both sides.
Divide by 3.
26 = x
mB = 2x + 3 = 2(26) + 3 = 55°
Holt McDougal Geometry
Angle Relationships in Triangles
Example 3
Find mACD.
mACD = mA + mB
Ext.  Thm.
6z – 9 = 2z + 1 + 90
Substitute 6z – 9 for mACD,
2z + 1 for mA, and 90 for mB.
6z – 9 = 2z + 91
Simplify.
4z = 100
Subtract 2z and add 9 to both
sides.
Divide by 4.
z = 25
mACD = 6z – 9 = 6(25) – 9 = 141°
Holt McDougal Geometry
Angle Relationships in Triangles
Example 4
Find mP and mT.
P  T
mP = mT
Third s Thm.
Def. of  s.
2x2 = 4x2 – 32 Substitute 2x2 for mP and 4x2 – 32 for mT.
–2x2 = –32
x2 = 16
Subtract 4x2 from both sides.
Divide both sides by -2.
So mP = 2x2 = 2(16) = 32°.
Since mP = mT, mT = 32°.
Holt McDougal Geometry
Angle Relationships in Triangles
Geometric figures are congruent if they are
the same size and shape. Corresponding
angles and corresponding sides are in the
same position in polygons with an equal
number of sides.
Two polygons are congruent polygons if
and only if their corresponding sides are
congruent. Thus triangles that are the same
size and shape are congruent.
Holt McDougal Geometry
Angle Relationships in Triangles
Holt McDougal Geometry
Angle Relationships in Triangles
Example 1
If polygon LMNP  polygon EFGH, identify all
pairs of corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH
Holt McDougal Geometry
Angle Relationships in Triangles
Example 2a
Given: ∆ABC  ∆DEF
a.) Find the value of x.
b.) Find mF.
AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
2x – 2 = 6
2x = 8
x=4
Holt McDougal Geometry
Substitute values for AB and DE.
Add 2 to both sides.
Divide both sides by 2.