Honors Geometry

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Honors Geometry

• Lesson 4.6
• Isosceles Triangles and Right Triangles

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What You Should LearnWhy You Should Learn It

• Goal 1 How to use properties of isosceles
  triangles to solve problems in geometry
• Goal 2 How to use properties of right triangles
  to solve problems in geometry
• You can use the properties of special triangles
  to solve problems related to solid figures with
  triangular faces, such as the equilateral
  triangles of an icosahedron (Example 1 on page
  197)

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Properties of Isosceles Triangles
Leg
Leg
Baseangles
Base
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Investigating Isosceles Triangles

• Use a straightedge and compass to construct an
  acute isosceles triangle
• Use scissors to cut the triangle out
• Then fold the triangle as shown
• Repeat the procedure for an obtuse isosceles
  triangle
• What observations can you make about the base
  angles of the triangle?

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Theorem 4.8 Base Angles Theorem

• If two sides of a triangle are congruent, then
  the angles opposite them are congruent
• Prove Theorem 4.8
• Given
• Prove

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Proof of Base Angles Theorem

• Given Prove

• Statements
• Label H as the midpoint of CY
• Draw NH

• Reasons
• Ruler Postulate
• 2 points determine a line
• Def. of midpoint
• Reflexive Prop
• Given
• SSS
• CPCTC

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Find the missing measures(not drawn to scale)

• 1.

• 2.

44
?
?
30
?
?
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Find the missing measures(not drawn to scale)

• 1.

• The two base angles are to each other b/c they
  are opposite congruent sides
• 180 44 136
• 136/2 68

44
68
68
?
?
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Find the missing measures(not drawn to scale)

• 2.

?
?
30
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Find the missing measures(not drawn to scale)

• The other base angle must be 30 b/c its opposite
  from a congruent side
• 180 (3030) 120

• 2.

?
120
?
30
30
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Theorem 4.9 (converse of Theorem 4.8)

• If two angles of a triangle are congruent, then
  the sides opposite them are congruent

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Corollary

• A corollary is a theorem that follows easily from
  a theorem that has been proven already
• Corollary 4.8 If triangle is equilateral, then
  it is also equiangular
• Corollary 4.9 If a triangle is equiangular, then
  it is also equilateral

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Theorem 4.10 Hypotenuse-Leg (HL) Congruence
Theorem

• If the hypotenuse and a leg of a right triangle
  are congruent to the hypotenuse and leg of a
  second right triangle, then the two triangles are
  congruent

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Proving Right Triangles Congruent

• The TV antenna is perpendicular to the plane
  containing the points B, C, D, and O. Each of the
  stays running from the top of the antenna to B,
  C, and D used the same length of cable. Is this
  enough information to conclude that ?AOB, ?AOC,
  and ?AOD are congruent?

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Proving Right Triangles Congruent

• B/C the antenna is perpendicular to ground level,
  you know that ?AOB, ?AOC, and ?AOD are _____
  triangles
• Each of these triangles share AO as a common leg
• B/C each stay uses the same length of cable, you
  can conclude that ____
• Apply the ________________________ Theorem to
  conclude that ?AOB, ?AOC, and ?AOD are congruent

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Proving Right Triangles Congruent

• B/C the antenna is perpendicular to ground level,
  you know that ?AOB, ?AOC, and ?AOD are right
  triangles
• Each of these triangles share AO as a common leg
• B/C each stay uses the same length of cable, you
  can conclude that
• Apply the ___________________Theorem to conclude
  that ?AOB, ?AOC, and ?AOD are congruent

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Proving Right Triangles Congruent

• B/C the antenna is perpendicular to ground level,
  you know that ?AOB, ?AOC, and ?AOD are right
  triangles
• Each of these triangles share AO as a common leg
• B/C each stay uses the same length of cable, you
  can conclude that
• Apply the ________________________ Theorem to
  conclude that ?AOB, ?AOC, and ?AOD are congruent

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Proving Right Triangles Congruent

• B/C the antenna is perpendicular to ground level,
  you know that ?AOB, ?AOC, and ?AOD are right
  triangles
• Each of these triangles share AO as a common leg
• B/C each stay uses the same length of cable, you
  can conclude that ____
• Apply the Hypotenuse-Leg Congruence Theorem
  Theorem to conclude that ?AOB, ?AOC, and ?AOD are
  congruent

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THE END