what was your goal? what was your actual grade? why did you meet/not meet your goal? what were...
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What was your goal? What was your actual grade? Why did you meet/not meet your goal? What were your strengths? What areas do you need to work on? What are you going to do to succeed
on the next test?
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Angles of a Triangle
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Write down everything you remember about triangles!
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By Side:› Equilateral—all sides congruent› Isosceles—two sides congruent› Scalene—no sides congruent
By Angle:› Obtuse—one angle greater than 90°› Right—one angle equal to 90°› Acute—all angles less than 90°
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Interior Angles add up to 180° Exterior Angles add up to 360° Examples:
› In triangle DEF; ∠D = 45°, ∠E = 30°. Find ∠F.
› In triangle ABC; ∠C = 3x – 5, ∠B = x + 40 and ∠A = 2x + 25. Find the measures of all three angles. 2x + 5
3x – 5 x + 40
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CPCTC and SSS
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Congruent line segments are marked with a small dash
Congruent angles are marked with an arc
Parallel lines are marked with arrows
To separate different pairs of congruent line segments or angles, we use different numbers of dashes or arcs
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Corresponding = matching Order of the letters matters! Example: ΔABC = ΔDEF. Which angles are
congruent? Which sides are congruent?
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SSS Postulate: If all three corresponding sides of two triangles are congruent, then the triangles are congruent
Example: ΔFEG = ΔKJL because of SSS.
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SAS, ASA, AAS, HL
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Name all the corresponding angles if ΔIJH ≅ ΔKJL.
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∠SRU ≅ ∠STU ∠RSU ≅ ∠TSU ∠RUS ≅ ∠TUS
Yes, can be proven through SSS
∠BRD ≅ ∠DYB ∠RBD ≅ ∠YDB ∠RDB ≅ ∠YBD
BR ≅ DY BY ≅ DR BD ≅ BD
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Side-Side-Side (SSS)
Side-Angle-Side (SAS)› Sandwich!
Angle-Side-Angle (ASA)› Sandwich!
Angle-Angle-Side (AAS)› No sandwich!
Hypotenuse-Leg (HL)› Right triangles only!
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Identity Properties in Triangle Proofs
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Reflexive Property: AB ≅ AB (congruent to itself)
Transitive Property: AB ≅ BC, BC ≅ CD, so AB ≅ CD
Additive Property: Adding the same amount to two congruent parts results in two equal sums
Multiplicative Property: Multiplying two congruent parts by the same number results in two equal products
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1. Mark diagram with “Given” and write as Step 1.2. Figure out how many parts of the triangles you
know are congruent, and how many you need to prove congruent.
3. Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns.
4. Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL.
5. Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.
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Given: GJ ≅ JIHJ ┴ GI Prove: ΔGJH ≅ ΔIJH
Statement Reason
1. GJ ≅ JIHJ ┴ GI
1. Given
2. ∠GJH ≅ ∠IJH2. ┴ lines form right angles, all right angles are ≅
3. HJ ≅ HJ 3. Reflexive property
4. ΔGJH ≅ ΔIJH 4. SAS
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Line/Angle Theorems in Triangle Proofs
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Midpoint› Halfway point on a line
segment Bisect
› Split a line segment or angle into two equal parts
HJ bisects GI
V is the midpoint of TW
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Vertical Angles › ALWAYS congruent; (“X”)
Alternate Interior Angles› ONLY congruent when we know lines are
parallel (“Z”)› ABCD is a
parallelogram
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1. Mark diagram with “Given” and write as Step 1.2. Figure out how many parts of the triangles you
know are congruent, and how many you need to prove congruent.
3. Mark missing congruent parts on diagram, using info from theorems you know (vertical angles, etc.). Write these down in the two columns.
4. Prove triangles congruent using: SSS, SAS, ASA, AAS, or HL.
5. Check: Make sure you used all info in the “Given.” Make sure your last step matches the “Prove”.
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Given: HK bisects IL ∠IHJ ≅ ∠JKL. Prove: ΔIHJ ≅ ΔLKJ
Statement Reason
1. HK bisects IL∠IHJ = ∠JKL
1. Given
2. IJ ≅ JL 2. Definition of “bisect”
3. ∠IJH ≅ ∠LJK 3. Vertical angles congruent
4. ΔIHJ ≅ ΔLKJ 4. AAS
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Using Quadrilateral Theorems in Triangle Proofs
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Parallelogram Rhombus Rectangle Square
ALSO WATCH OUT FOR:› Alternate Interior Angles› Vertical Angles
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• Opposite sides are parallel and congruent
• •
• Opposite angles are congruent
• • Diagonals bisect each other
• Bisect = to split in half
CBADCDAB ,
CBADCDAB ,
CDAABCBCDDAB ,
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• Has all the properties of a parallelogram, plus:• FOUR congruent
sides• Diagonals are
perpendicular and bisect
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• Has all properties of a parallelogram, plus:• Four right angles• Congruent
diagonals that bisect
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• Four congruent sides and four right angles
• Diagonals are congruent and perpendicular; also bisect
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Given: FLSH is a parallelogram; LG ┴ FS, AH ┴ FS
Prove: ΔLGS ≅ ΔHAFStatement Reason
1. FLSH is a parallelogram; LG ┴ FS, AH ┴ FS
1. Given
2. LG ≅ FH 2. Opp. sides of p.gram are ≅
3. ∠LGS ≅ ∠HAF 3. ┴ lines form right ∠’s, all right ∠’s ≅
4. ∠LSG ≅ ∠HFA 4. Alt. int. ∠’s ≅ when lines ||
5. ΔLGS ≅ ΔHAF 5. AAS
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Using Circle Theorems in Triangle Proofs
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Chords intercepting congruent arcs are congruent
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Tangent is perpendicular to the radius at the point where it touches the circle
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Arcs between parallel lines are congruent
H
F
J
G
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Inscribed angle is half the intercepted arc.
Two inscribed angles that intercept the same arc are congruent
N
J
LM
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Given: arc BR = 70°, arc YD = 70°; BD is the diameter of circle O
Prove: ΔRBD ≅ ΔYDBStatement Reason
1. Arc BR = 70°, arc YD = 70°; BD is the diam. of circle O
1. Given
2. BD ≅ BD 2. Reflexive
3. ∠YBD = 35°, ∠RDB = 35°
3. Inscribed angles = ½ arc
4. ∠YBD ≅ ∠RDB 4. ≅ arcs have same measure
5. ∠BYD ≅ ∠BRD5. Inscribed angles intercepting same arc are ≅
5. ΔRBD ≅ ΔYDB 6. AAS