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4-1
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4-1 Triangles and Angles
Holt Geometry
Warm UpWarm UpLesson PresentationLesson PresentationLesson QuizLesson Quiz
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Warm UpClassify each angle as acute, obtuse, or right.
1. 2.
3.
4. If the perimeter is 47, find x and the lengths of the three sides.
right acute
x = 5; 8; 16; 23
obtuse
4.1 Triangles and Angles
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Classify triangles by their angle measures and side lengths.Use triangle classification to find angle measures and side lengths. Find the measures of interior and exterior angles of triangles.Apply theorems about the interior and exterior angles of triangles.
Objectives
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acute triangle Corollaryequiangular triangle Legsright triangle Adjacentobtuse triangle Exteriorequilateral triangle Interiorisosceles triangle Hypotenusescalene triangle base
Vocabulary
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Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.
4.1 Triangles and Angles
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B
AC
AB, BC, and AC are the sides of ABC.A, B, C are the triangle's vertices.
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4.1 Triangles and AnglesIn a triangle, two sides sharing a commonvertex are Adjacent sides.
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4.1 Triangles and AnglesIn a right triangle, the two sides making the right angle are called Legs.
The side opposite of the right angle is called the Hypothenuse.
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4.1 Triangles and AnglesIn an isosceles triangle, the two sides that are congruent are called the Legs.
The third side is the called the base.
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Acute Triangle
Three acute angles
Triangle Classification By Angle Measures
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Equiangular Triangle
Three congruent acute angles
Triangle Classification By Angle Measures
4.1 Triangles and Angles
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Right Triangle
One right angle
Triangle Classification By Angle Measures
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Obtuse Triangle
One obtuse angle
Triangle Classification By Angle Measures
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Classify BDC by its angle measures.
Example 1A: Classifying Triangles by Angle Measures
B is an obtuse angle. B is an obtuse angle. So BDC is an obtuse triangle.
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Classify ABD by its angle measures.
Example 1B: Classifying Triangles by Angle Measures
ABD and CBD form a linear pair, so they are supplementary. Therefore mABD + mCBD = 180°. By substitution, mABD + 100° = 180°. So mABD = 80°. ABD is an acute triangle by definition.
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Classify FHG by its angle measures.
Check It Out! Example 1
EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°.
FHG is an equiangular triangle by definition.
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Equilateral Triangle
Three congruent sides
Triangle Classification By Side Lengths
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Isosceles Triangle
At least two congruent sides
Triangle Classification By Side Lengths
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Scalene Triangle
No congruent sides
Triangle Classification By Side Lengths
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Remember!When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.
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4.1 Triangles and Angles
When the sides of the triangle are extended, the original angles are the interior angles.
The angles that are adjacent (next to) the interior angles are the exterior angles.
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4.1 Triangles and Angles
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An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum
Theorem
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A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.
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An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum
Theorem
4.1 Triangles and Angles
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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 mXYZ.
Example 1A: Application
mXYZ + mYZX + mZXY = 180° Sum. Thm
mXYZ + 40 + 62 = 180 Substitute 40 for mYZX and 62 for mZXY.
mXYZ + 102 = 180 Simplify.
mXYZ = 78° Subtract 102 from both sides.
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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 mYWZ.
Example 1B: Application
mYXZ + mWXY = 180° Lin. Pair Thm. and Add. Post.
62 + mWXY = 180 Substitute 62 for mYXZ.
mWXY = 118° Subtract 62 from both sides.
Step 1 Find mWXY. 118°
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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 mYWZ.
Example 1B: Application Continued
Step 2 Find mYWZ. 118°
mYWX + mWXY + mXYW = 180° Sum. Thm
mYWX + 118 + 12 = 180 Substitute 118 for mWXY and 12 for mXYW.
mYWX + 130 = 180 Simplify. mYWX = 50° Subtract 130 from both sides.
4.2 Congruence and Triangles
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Use the diagram to find mMJK.
Check It Out! Example 1
mMJK + mJKM + mKMJ = 180° Sum. Thm
mMJK + 104 + 44= 180 Substitute 104 for mJKM and 44 for mKMJ.
mMJK + 148 = 180 Simplify. mMJK = 32° Subtract 148 from both sides.
4.1 Triangles and Angles
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A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.
4.1 Triangles and Angles
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One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?
Example 2: Finding Angle Measures in Right Triangles
mA + mB = 90°2x + mB = 90 Substitute 2x for mA.
mB = (90 – 2x)° Subtract 2x from both sides.
Let the acute angles be A and B, with mA = 2x°.
Acute s of rt. are comp.
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The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?
Check It Out! Example 2a
mA + mB = 90°63.7 + mB = 90 Substitute 63.7 for mA.
mB = 26.3° Subtract 63.7 from both sides.
Let the acute angles be A and B, with mA = 63.7°.
Acute s of rt. are comp.
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The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?
Check It Out! Example 2b
mA + mB = 90°x + mB = 90 Substitute x for mA.
mB = (90 – x)° Subtract x from both sides.
Let the acute angles be A and B, with mA = x°.
Acute s of rt. are comp.
4.1 Triangles and Angles
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The measure of one of the acute angles in a right triangle is 48 . What is the measure of the other acute angle?
Check It Out! Example 2c
mA + mB = 90° Acute s of rt. are comp.
2° 5
Let the acute angles be A and B, with mA = 48 . 2° 5
Subtract 48 from both sides. 2 5
Substitute 48 for mA. 2 548 + mB = 902
5mB = 41 3°
5
4.2 Congruence and Triangles
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The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.
Interior
Exterior
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An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side.
Interior
Exterior4 is an exterior angle.
3 is an interior angle.
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Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.
Interior
Exterior
3 is an interior angle.
4 is an exterior angle. The remote interior angles of 4 are 1 and 2.
4.1 Triangles and Angles
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Find mB.Example 3: Applying the Exterior Angle Theorem
mA + mB = mBCD Ext. Thm.
15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
2x + 18 = 5x – 60 Simplify.
78 = 3x Subtract 2x and add 60 to both sides.
26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55°
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Find mACD.Check It Out! Example 3
mACD = mA + mB Ext. Thm.
6z – 9 = 2z + 1 + 90 Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.
6z – 9 = 2z + 91 Simplify.
4z = 100 Subtract 2z and add 9 to both sides.
z = 25 Divide by 4. mACD = 6z – 9 = 6(25) – 9 = 141°
4.2 Congruence and Triangles
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Classify EHF by its side lengths.
Example 2A: Classifying Triangles by Side Lengths
From the figure, . So HF = 10, and EHF is isosceles.
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Classify EHG by its side lengths.
Example 2B: Classifying Triangles by Side Lengths
By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene.
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Classify ACD by its side lengths.
Check It Out! Example 2
From the figure, . So AC = 15, and ACD is isosceles.
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Find the side lengths of JKL.
Example 3: Using Triangle Classification
Step 1 Find the value of x. Given.
JK = KL Def. of segs.
4x – 10.7 = 2x + 6.3 Substitute (4x – 10.7) for JK and (2x + 6.3) for KL.
2x = 17.0
x = 8.5
Add 10.7 and subtract 2x from both sides.Divide both sides by 2.
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Find the side lengths of JKL.
Example 3 Continued
Step 2 Substitute 8.5 into the expressions to find the side lengths. JK = 4x – 10.7
= 4(8.5) – 10.7 = 23.3KL = 2x + 6.3
= 2(8.5) + 6.3 = 23.3JL = 5x + 2
= 5(8.5) + 2 = 44.5
4.1 Triangles and Angles
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Find the side lengths of equilateral FGH.
Check It Out! Example 3
Step 1 Find the value of y.
Given.FG = GH = FH Def. of segs.
3y – 4 = 2y + 3Substitute (3y – 4) for FG and (2y + 3) for GH.
y = 7 Add 4 and subtract 2y from both sides.
4.1 Triangles and Angles
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Find the side lengths of equilateral FGH.
Check It Out! Example 3 Continued
Step 2 Substitute 7 into the expressions to find the side lengths.
FG = 3y – 4 = 3(7) – 4 = 17
GH = 2y + 3 = 2(7) + 3 = 17
FH = 5y – 18 = 5(7) – 18 = 17
4.1 Triangles and Angles
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The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.P = 3(18)P = 54 ft
A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
Example 4: Application
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A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
Example 4: Application Continued
To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle.420 54 = 7 triangles 7
9There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.
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The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.P = 3(7)P = 21 in.
Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.
Check It Out! Example 4a
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To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.100 7 = 14 triangles 2
7
There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel.
Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.
Check It Out! Example 4a Continued
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The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.P = 3(10)P = 30 in.
Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.
Check It Out! Example 4b
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To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.100 10 = 10 triangles
The manufacturer can make 10 triangles from a 100 in. piece of steel.
Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.
Check It Out! Example 4b Continued
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Lesson Quiz IClassify each triangle by its angles and sides.
1. MNQ 2. NQP 3. MNP
4. Find the side lengths of the triangle.
acute; equilateral
obtuse; scalene
acute; scalene
29; 29; 23
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Lesson Quiz: Part II
5. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle?
6. Find mABD.
124°
2 3
33 °1 3
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Lesson Quiz: Part III7. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?
30°
4.1 Triangles and Angles