module’9’ properes’of’ quadrilaterals’ part1 · 8/15/2019 · definion • a...
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Module 9 Proper,es of Quadrilaterals
Part 1: Parallelograms
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Defini,on • A parallelogram is a quadrilateral whose opposite sides are parallel.
• Its symbol is a small figure:
CB
A D
AB CD and BC AD
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Naming a Parallelogram
• A parallelogram is named using all four ver,ces.
• You can start from any one vertex, but you must con,nue in a clockwise or counterclockwise direc,on.
• For example, this can be either ABCD or ADCB. CB
A D4
Basic Proper,es • There are four basic proper,es of all parallelograms. – Opposite Sides – Opposite Angles – Consecu,ve Angles – Diagonals
• These proper,es have to do with the angles, the sides and the diagonals.
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Opposite Sides Theorem Opposite sides of a parallelogram are congruent.
• That means that . • So, if AB = 7, then _____ = 7?
CB
A D
AB≅CD and BC ≅ ADCD
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Opposite Angles One pair of opposite angles is ∠A and ∠ C. The other pair is ∠ B and ∠ D.
Theorem Opposite angles of a parallelogram are congruent.
• Complete: If m ∠ A = 75° and m ∠ B = 105°, then m ∠ C = ______ and m ∠ D = ______ .
CB
A D8
75° 105°
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Consecu,ve Angles • Each angle is consecu,ve to two other angles. ∠A is consecu,ve with ∠ B and ∠ D.
CB
A D
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Consecu,ve Angles in Parallelograms
Theorem Consecu,ve angles in a parallelogram are supplementary.
• Therefore, m ∠ A + m ∠ B = 180° and m ∠ A + m ∠ D = 180°.
• If m<C = 46°, then m ∠ B = _____?
CB
A D
Consecutive INTERIOR Angles are
Supplementary!
134°
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Diagonals • Diagonals are segments that join non-‐consecu,ve ver,ces.
• For example, in this diagram, the only two diagonals are . AC and BD
CB
A D
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Diagonal Property When the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal.
• So, P is the midpoint of . • Therefore, they bisect each other;
so and . • But, the diagonals are not congruent!
AC and BD
AP ≅ PC BP≅ PD
P
CB
A D
AC ≠ BD
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Diagonal Property Theorem The diagonals of a parallelogram bisect each other.
P
CB
A D
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Parallelogram Summary • By its defini,on, opposite sides are parallel. Other proper,es (theorems): • Opposite sides are congruent. • Opposite angles are congruent. • Consecu,ve angles are supplementary. • The diagonals bisect each other.
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Examples
• 1. Draw HKLP. • 2. Complete: HK = _______ and
HP = ________ . • 3. m<K = m<______ . • 4. m<L + m<______ = 180°. • 5. If m<P = 65°, then m<H = ____,
m<K = ______ and m<L =______ .
PL KL
P
P or <K
115° 65° 115°
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Examples (cont’d)
• 6. Draw in the diagonals. They intersect at M.
• 7. Complete: If HM = 5, then ML = ____ . • 8. If KM = 7, then KP = ____ . • 9. If HL = 15, then ML = ____ . • 10. If m<HPK = 36°, then m<PKL = _____ .
5
14
7.5
36°
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Part 2
Tests for Parallelograms
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Review: Proper,es of Parallelograms
• Opposite sides are parallel. • Opposite sides are congruent. • Opposite angles are congruent. • Consecu,ve angles are supplementary. • The diagonals bisect each other.
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How can you tell if a quadrilateral is a parallelogram?
• Defn: A quadrilateral is a parallelogram iff opposite sides are parallel.
• Property If a quadrilateral is a parallelogram, then opposite sides are parallel.
• Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram.
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Proving Quadrilaterals as Parallelograms
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .
Theorem 1:
H G
E F
If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .
Theorem 2:
If EF GH; FG EH, then Quad. EFGH is a parallelogram.≅ ≅
If EF GH and EF || HG, then Quad. EFGH is a parallelogram.≅24
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If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 3:
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .
Theorem 4:
H G
E F
M
,If H F and E G∠ ≅ ∠ ∠ ≅ ∠
then Quad. EFGH is a parallelogram.
intIf M is themidpo of EG and FHthen Quad. EFGH is a parallelogram. EM = GM and HM = FM
Proving Quadrilaterals as Parallelograms (part 2)
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5 ways to prove that a quadrilateral is a parallelogram.
1. Show that both pairs of opposite sides are || . [definition]
2. Show that both pairs of opposite sides are ≅ .
3. Show that one pair of opposite sides are both || and ≅ .
4. Show that both pairs of opposite angles are ≅ .
5. Show that the diagonals bisect each other .
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Examples …… Find the values of x and y that ensures the quadrilateral is a parallelogram.
Example 1:
6x 4x+8
y+2
2y
6x = 4x + 8
2x = 8
x = 4
2y = y + 2
y = 2
Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.
120°
5y° (2x + 8)° 2x + 8 = 120
2x = 112
x = 56
5y + 120 = 180
5y = 60
y = 12 29
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9.1-‐9.2 Classwork PAGE 426
• GO ONLINE and complete 9.1-‐9.2 hw. • AlternaEve: Honors: 9.1: 3, 5-‐6, 14, 17-‐18, 23-‐24
9.2: 1, 5, 8, 11-‐12, 18-‐19 • Regular: 9.1: 5-‐6, 8, 17-‐18 9.2: 1, 5, 8, 12, 18 Reminders: q …
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