Midline Theorems, Trapezoid and Kites PDF

Preview

Summary

Lecture notes on midline theorems, trapezoid and kites...

Description

LAS on Mathematics 9 Week 3- Midline Theorems, Trapezoid and Kites Midline Theorem – The segment that joins the midpoints of two sides of a triangle is parallel to the third side and half as long.

Given: ∆HNS, O is the midpoint of HN and E is the midpoint of NS Prove: OE//HS and OE=12HS Statements Reasons 1. ∆HNS, O is the midpoint of HN and E is the midpoint of Given NS 2. In OE, there is a point T such that OE≅ET Line Postulate Definition of a midpoint 3. NE≅ES 4. ∠2≅∠3 Vertical Angles Theorem 5. ∆NEO≅∆SET SAS Postulate 6. ∠1≅∠4 CPCTC 7. HN//ST Converse of Alternate Interior Angles Theorem 8. OH≅ON Definition of a midpoint 9. ON≅TS CPCTC 10. OH≅TS Transitive Property 11. Quadrilateral HOTS is a parallelogram If opposite sides of a quadrilateral are congruent and parallel, then it is a parallelogram. 12. OE//HS Definition of a parallelogram Segment Addition Postulate 13. OE+ET=OT 14. OE+OE=OT Substitution Property 15. 2OE=OT Addition Property 16. HS≅OT In a parallelogram, any two opposite sides are congruent. 17. 2OE=HS Substitution Property 18. OE=12HS Divide both sides by two Definition of Trapezoid – a quadrilateral with one pair of parallel sides. Midsegment Theorem of Trapezoid – The median of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. Given: Trapezoid MINS with median TR that intersects the diagonal IS at P. Prove: MS//TR//IN and TR=12(MS+IN)

Statements 1. Trapezoid MINS with median TR that intersects the diagonal IS at P. 2. TP+PR=TR 3. TP//MS and TP=12MS RP//IN and RP=12IN 4. MS//IN 5. MS//TP+RP//IN 6. MS//TR//IN 7. TR=12MS+12IN 8. TR=12(MS+IN)

Reasons Given Segment Addition Postulate Midline Theorem Definition of Trapezoid Transitive Property Addition Property Substitution Property Factoring

Definition of Isosceles Trapezoid – a trapezoid with opposite sides that are congruent. Properties of Isosceles Trapezoid

• The base angles of an isosceles trapezoid are congruent. Given: Isosceles Trapezoid AMOR with MO//AR Prove: ∠A≅∠R and ∠AMO≅∠O Statements 1. Isosceles Trapezoid AMOR with MO//AR. 2. AM≅OR 3. From M, draw ME//OR where E lies on AR 4. Quadrilateral MORE is a parallelogram 5. ME≅OR 6. 7. 8. 9. 10. 11. angles.

MA≅ME ∆AME is an isosceles triangle ∠A≅∠1 ∠1≅∠R ∠A≅∠R ∠A and ∠AMO are supplementary angles. ∠O and ∠R are supplementary

Reasons Given Definition of Isosceles Trapezoid Two points determine a line Definition of Parallelogram In a parallelogram, any two opposite sides are congruent. Transitive Property Definition of Isosceles Triangle Base angles of an isosceles triangles are congruent. Corresponding Angles Theorem Transitive Property Same Side Interior Angle Theorem

12.

• •

Supplements of congruent angles are also congruent

∠AMO≅∠O

Opposite angles of an isosceles trapezoid are supplementary. Diagonals of an isosceles trapezoid are congruent.

Definition of Kite – a quadrilateral with two pairs of adjacent sides that are congruent, a rhombus is a special kind of kite. Properties of Kite • In a kite, the perpendicular bisector of at least one is the other diagonal. • The area of a kite is half the product of the lengths of its diagonals. Given: Kite ROPE with diagonals PR and OE intersect at point W. Prove: Area of kite ROPE = 12(UE)(PR).

Statements 1. Kite ROPE with diagonals PR and OE intersect at point W. 2. PROE 3. Area of kite ROPE = Area of ∆OPE + Area of ∆ORE 4. Area of ∆OPE = 12(OE)(PW) and Area of ∆ORE = 12(OE)(WR) 5. Area of kite ROPE = =12(OE)(PW+WR) 6. PW+WR=PR 7. Area of Kite ROPE = 12(UE)(PR) 8. CX≅TX

Reasons Given Diagonals of a kite are perpendicular to each other. Area Addition Postulate Formula for Area of Triangles Factoring Segment Addition Postulate Substitution CPCTC

Learning Activity 1: Directions: Complete the two-column proof. • Opposite angles of an isosceles trapezoid are supplementary. Given: Isosceles Trapezoid ARTS with RT//AS Statements Prove: ∠ARS and ∠S are supplementary. ∠A and ∠T are 1. (1) supplementary. 2. AR≅ST 3. (3) and ∠ART≅∠S 4. ∠A+∠ART=1800 and ∠S+∠T=1800 5. ∠A+∠T=1800 and ∠S+∠ART=1800 6. (5)

Reasons Given (2) The base angles of an isosceles trapezoid are congruent. Same Side Interior Angle Theorem (4) Definition of Supplementary Angles

Learning Activity 2: Directions: Complete the two-column proof. • Diagonals of an isosceles trapezoid are congruent. Given: Isosceles Trapezoid ROMA with diagonals RM Statements and AO 1. Isosceles Trapezoid ROMA with diagonals RM and AO. 2. (2)

Prove: RM≅AO

3. 4. 5. 6.

∠ORA≅∠MAR RA≅AR (5) RM≅AO

Reasons (1) Definition of Isosceles Trapezoid (3) (4) SAS Postulate CPCTC

Learning Activity 3: Directions: Complete the two-column proof. • A diagonal of a kite is an angle bisector of a pair of opposite angles. Given: Kite WORD with diagonals OD and WR. Statements Prove: WR is angle bisector of ∠OWD and ∠ORD. 1. Kite WORD with diagonals OD and WR. 2. WO≅WD and RO≅RD 3. WR≅WR 4. (3) 5. (4) 6. WR is angle bisector of ∠OWD and ∠ORD.

Reasons (1) (2) Reflexive Property SSS Postulate (5) Definition of Angle Bisector

LeaP_Mathematics_Wilson Ray G. Anzures - SDO San Pablo City...