Grade 9 MATH Term 1 Week 8: Properties of Parallelograms to Find Missing Measures MATATAG Quarter 2

Main ideas and lessons (about parallelograms in Grade 9 Math)

The video teaches how to use parallelogram properties to find missing angle measures, side lengths, and diagonal segments, as well as perpendicular height using area.

It first reviews what parallelograms are and how to identify them among other quadrilaterals, then lists key properties, and finally demonstrates step-by-step example problems.

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Review: Quadrilateral and parallelogram identification

• Quadrilateral definition: a simple closed polygon with:
  • 4 sides
  • 4 vertices
  • 4 angles

Which shapes are parallelograms (from the quadrilateral family mentioned)

• Square ✅
• Rectangle ✅
• Rhombus ✅
• Isosceles trapezoid / trapezium (mentioned) ✅/included in “parallelogram classifications” in the video’s review context
• Kite ✅/included in the review context
  • (The wording is inconsistent in subtitles, but the intent is parallelogram vs non-parallelogram family.)

Core identification used: A parallelogram is a quadrilateral with two pairs of opposite parallel sides (i.e., opposite sides are parallel).

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Naming and parts of a parallelogram

Naming

• Name using four vertices in order:
  • Clockwise or counterclockwise, e.g., parallelogram ABCD.

Opposite sides

• Example labeling:
  • AB opposite CD
  • AD opposite BC

Parallel opposite sides

• Opposite sides are parallel:
  • AB ∥ CD
  • AD ∥ BC

Opposite angles and consecutive angles

• Opposite angles: (A with C), (B with D)
• Consecutive angles: adjacent angles around the parallelogram:
  • A & B, B & C, C & D, D & A

Diagonals

• Diagonals connect opposite vertices:
  • AC and BD (or equivalent depending on labeling)

Perpendicular height

• The perpendicular height is the distance between the two parallel bases, measured at a right angle.
• There are two possible perpendicular heights, depending on which side is treated as the base (the video conceptually demonstrates “two” cases).

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Properties of a parallelogram (main rules used to solve problems)

1. Opposite sides are congruent (equal) Both pairs of opposite sides have the same length.

2. Opposite angles are congruent Two opposite angles have equal measures.

3. Consecutive angles are supplementary Any two adjacent angles add to 180°.

4. Diagonals bisect each other The diagonals intersect and cut each other into equal halves.

   • If diagonals intersect at U, then LU = UV and EU = UO (segment equalities).

5. A diagonal divides a parallelogram into two congruent triangles The diagonal splits the parallelogram into two congruent triangles.

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Methodology shown in examples (step-by-step approach)

A) Finding missing side lengths or diagonal parts

• Use Property #1 (opposite sides congruent):
  • If one side is given, the opposite side equals it.
• Use Property #4 (diagonals bisect each other):
  • If a diagonal segment is given as a total (e.g., XC = 50), then each half is:
    • (1/2) of the diagonal
    • missing segment = diagonal ÷ 2

B) Finding missing angle measures

• Use Property #2 (opposite angles congruent):
  • Missing opposite angle = given angle.
• Use Property #3 (consecutive angles supplementary):
  • Missing adjacent angle = 180° − given angle
• With algebraic angle expressions:
  • Set up equations using:
    • consecutive angles sum to 180°
  • Substitute expressions, solve for the variable, then substitute back to get exact angle values.

C) Finding perpendicular height using area

• Use the parallelogram area relationship:
  • Area = base × height
• Solve for height:
  • height = area ÷ base
• Substitute given measurements to compute height.

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Examples/worked tasks demonstrated

Example 1: Parallelogram WXYZ

• Given:
  • XY = 36
  • XC (diagonal segment) = 50
  • ∠WXY = 68°
• Asked:
  • Find WZ (a diagonal midpoint segment) and missing angles.
• Key solutions used:
  • Opposite sides congruent → WZ = XY = 36
  • Diagonals bisect each other → midpoint segment = 50 ÷ 2 = 25
  • Opposite angles congruent → ∠YZW = ∠WXY = 68°
  • Consecutive angles supplementary → ∠ZWX = 180° − 68° = 112°

Example 2: Parallelogram DEFG (angle given)

• Given: ∠D = 58°
• Asked: remaining angles (E, F, G)
• Key solutions used:
  • Opposite angles congruent → ∠F = 58°
  • Consecutive angles supplementary → ∠E = 180° − 58° = 122°
  • Opposite angles congruent again → ∠G = ∠E = 122°

Example 3: Perpendicular height from area

Asked for perpendicular height given (two cases):

• (A) area = 180 m², base = 20 m → height = 180 ÷ 20 = 9 m
• (B) area = 252 m², base = 18 m → height = 252 ÷ 18 = 14 m

Example 4: Solve for angles using algebra

• Two consecutive angles:
  • One angle = X + 35
  • Adjacent angle = 2X − 5
• Use consecutive angles supplementary:
  • (X + 35) + (2X − 5) = 180
  • 3X + 30 = 180
  • 3X = 150
  • X = 50
• Then:
  • ∠K = X + 35 = 85°
  • ∠N = 2X − 5 = 95°
• Opposite angles congruent:
  • ∠M = ∠K = 85°
  • ∠L = ∠N = 95°

Example 5: Solve for M and N using sides + angles

• Given:
  • Side WI = 3M + 7
  • Opposite side ES = 28
  • Angle IWS = 3N
  • Angle ∠I = 115°
  • Angle ∠SWE = 20°
• For M:
  • Opposite sides congruent → 3M + 7 = 28 → M = 7
• For N:
  • Angle addition: ∠IWE = ∠IWS + ∠SWE
    • ∠IWE = 3N + 20
  • Consecutive angles supplementary with ∠I:
    • (3N + 20) + 115 = 180
    • 3N + 135 = 180
    • 3N = 45
    • N = 15

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Final summary of the key rules (as emphasized at the end)

• Opposite sides are parallel and congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary (sum to 180°).
• Diagonals bisect each other.
• A diagonal divides a parallelogram into two congruent triangles.
• Perpendicular height uses:
  • Area = base × height, so height = area ÷ base.

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Speakers / sources featured

• Main speaker/teacher (unidentified by name): the instructor who presents the lesson, examples, and explanations.
• No other clearly identified speakers, guests, or sources were mentioned in the subtitles.