MATH Let:
Ratio and Proportion x = be the supplement of the angle
x+40 = measure of the angle
x + x + 40 = 180
Similarity 2x = 180 - 40
● it is the state or fact 2x = 140
of being similar x = 70
Ratio x + 40 = 70 + 40 = 110
● Ratio is a comparison by division of 110:70
two numbers denoted by :, /, -
● Given two numbers a & b where
b≠0, their ratio can be written in any Proportion
of the three following ways: ● a part, share, or number considered
1.
𝑎 in comparative relation to a whole
𝑏
● An equation in which two ratios are
2. a:b equal
3. a to b 𝑎 𝑐
𝑏
= 𝑑
𝑜𝑟 𝑎: 𝑏 = 𝑐: 𝑑
where b≠0, and d≠0 and a, b, c, & d
Examples are the terms
Mark and Luke obtained 36 and 45 points in Fundamental Law of Proportion:
their math quiz, respectively. Express the ● The product of the means is equal to
scores of Mark and Luke as a ratio. the product of the extremes
Mark - 36
Luke - 45
36 9 4
45
÷ 9
= 5
4:5
The measures of the two consecutive angles
of a parallelogram are in the ratio 5:7.
Let:
5x > Be the angles of parallelogram
7x
5x + 7x = 180
12x = 180
12𝑥 180
12
= 12
x = 15
5x = 5(15) = 75
7x = 7(15) = 105
75 + 105 = 180
Examples
Similar Triangles
● Similar triangles have congruent
corresponding angles and
proportional corresponding sides
● Triangles are similar if you show:
○ Any 2 pairs of corresponding
sides are proportional and
the included angles are
congruent (SAS Similarity)
All 3 pairs of corresponding sides are
proportional (SSS Similarity)
Proportion of Similar Triangles
Similar Polygons
● Similar polygons have:
○ Congruent
corresponding angles
○ Proportional
corresponding sides
● Scale factor: the ratio of
corresponding sides
Any 2 pairs of corresponding angles are ● If 2 triangles are
congruent (AA Similarity) similar, then the
measures of the
corresponding
medians are
proportional to the
corresponding sides.
If two triangles are similar, then the
perimeters are proportional to the measures
of corresponding sides
● An angle bisector in a
triangle cuts the
opposite side into
segments that are
proportional to the
other sides
Parts of Similar Triangles
● If two triangles are similar:
○ the measures of the
corresponding altitudes are
proportional to the
corresponding sides
Theorem
Triangle Proportionality Theorem
● If a line is parallel to one side of a
triangle and intersects the other two
sides, then it divides the two sides
○ the measures of the proportionally.
corresponding angle
bisectors are proportional to
the corresponding sides
Converse of the Triangle
Proportionality Theorem
● If a line divides two sides of a
triangle proportionally, then
the line is parallel to the
remaining side.
Special Right Triangles (30-60-90)
Special Right Triangles (45-45-90)
Leg to Hypotenuse: MULTIPLY by 2
Hypotenuse to Leg: DIVIDE by 2
The Pythagorean Theorem
Pythagoras Theorem (also called
Pythagorean Theorem)
● is an important topic in Mathematics,
which explains the relation between
the sides of a right-angled triangle.
● The sides of the right triangle are
also called Pythagorean triples.
● basically used to find the length of
an unknown side and the angle of a
triangle.
● By this theorem, we can derive the
base, perpendicular, and
hypotenuse formulas.
Pythagorean Theorem Statement
● “In a right-angled triangle, the
square of the hypotenuse side is
equal to the sum of squares of the
other two sides“.
The proof of Pythagorean Theorem
Shorter Leg to Hypotenuse: MULTIPLY by 2
Hypotenuse to Shorter Leg: DIVIDE by 2
Shorter Leg to Longer Leg: MULTIPLY by
3
Longer leg to Shorter Leg: DIVIDE by 3
Examples
Properties of Quadrilaterals
Quadrilateral
● a two-dimensional figure with four
sides and four angles.
● The word part “quad” means 4 and
“lateral” means sides.
Types of Quadrilaterals
Square
● a quadrilateral with 4 equal sides
and 4 right angles
Rectangle
● a quadrilateral with 4 right angles.
● Its opposite sides are equal and
parallel.
Is a square a rectangle?
● A square can also be called a
rectangle because it has four right
angles and its opposite sides are
equal and parallel.
Is a rectangle a square?
● A rectangle is NOT a square
because it does not have equal
sides.
Trapezoid
Properties of Parallelograms
● a quadrilateral that has exactly 1 pair
of parallel sides
● has one set of parallel sides Theorems about parallelograms
● If a quadrilateral is a parallelogram,
Rhombus then its opposite sides are
● a quadrilateral that has 4 equal sides congruent.
● Its opposite sides are parallel ○ PQ ≅ RS and SP ≅ QR
Parallelogram
● a quadrilateral that has opposite
sides that are equal and parallel
● If a quadrilateral is a parallelogram,
then its opposite angles are
congruent.
○ ∠P ≅ ∠R and ∠Q ≅ ∠S
● If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary (add up to 180°).
○ m∠P +m∠Q = 180°,
○ m∠Q +m∠R = 180°,
○ m∠R + m∠S = 180°,
○ m∠S + m∠P = 180°
● If a quadrilateral is a parallelogram,
then its diagonals bisect each other.
○ QM ≅ SM and PM ≅ RM
Right Triangles Using Trigonometric Ratios
● Sine - “sign”
● Cosine - “co-sign”
● Tangent - “tan-gent”
Greek Letter θ - “theta”
● Represents an unknown angle
Hypotenuse - always opposite the
right-angle & always longest
Opposite - always opposite θ
Adjacent - next to θ
Angles of Elevation & Depression
An angle of elevation is the angle formed
by a horizontal line and a line of sight to a
point above the line. In the diagram, ∠1 is
the angle of elevation from the tower T to
the plane P.
An angle of depression is the angle
formed by a horizontal line and a line of
sight to a point below the line. ∠2 is the
angle of depression from the plane to the
tower.
Since horizontal lines are parallel, ∠1 ≅ ∠2
by the Alternate Interior Angles Theorem.
Therefore the angle of elevation from one
point is congruent to the angle of
depression from the other point.
