Question: Activity 12.1
Angle relationships – perpendicular line and intersecting lines
Term 2
Week 7
20 minutes
1. Find the size of the unknown angle in each question. State which property you use:
a) 130° a
b) 42° b
c) 29° c 53°
d) 112° d
e) 121° x y
f) 18° f
g) 45° g 24°
h) 155° h
i) x 96° x
j) 23° 95° x
2. In the diagram below, the larger angle is four times bigger than the smaller angle.
Find the two unknown angles.
3. Find the size of x in each of the following:
a) 2x 4x
b) x x 2x
c) 5x 4x + 60
d) 120° x 2x
e) 78° 3x x 2x
f) x 3x
Graph shape and math:
Top-left: two intersecting lines form vertical opposite and supplementary angles; one angle is 130° and the adjacent angle is a, so a = 50°.
Top-right: two intersecting lines form vertical opposite angles; one angle is 42° and the opposite angle is b, so b = 42°.
Top-right: a straight line with a ray creates adjacent angles 29° and c, and another adjacent angle is 53°; c = 180° - 29° - 53° = 98°.
Center: two intersecting lines with a perpendicular line create angles labeled d and 112°; d = 68°.
Center: intersecting lines with angles 121°, x, and y around the vertex; x and y are supplementary/vertical depending on position.
Center: a right angle marker shows f = 90° - 18° = 72°.
Center-left: angles 45°, g, and 24° around an intersection; g = 180° - 45° - 24° = 111°.
Center-right: angles 19°, 155°, and h around intersecting lines; h is the remaining angle at the intersection.
Bottom-right: a straight line with two rays above it and a central angle 96°, with x on both sides; the two x angles are supplementary to 96° on the straight line.
Bottom-left: a single straight line with an oblique ray forms a larger and smaller angle for the "four times bigger" question.
1. **Find the size of the unknown angle in each question and state the property used:**
a) Given angle is $130^\circ$ and adjacent angle $a$ forms a straight line.
Property: Supplementary angles sum to $180^\circ$.
$$a = 180^\circ - 130^\circ = 50^\circ$$
b) Given angle is $42^\circ$ and $b$ is vertically opposite angle.
Property: Vertically opposite angles are equal.
$$b = 42^\circ$$
c) Angles $29^\circ$, $c$, and $53^\circ$ are adjacent on a straight line.
Property: Angles on a straight line sum to $180^\circ$.
$$c = 180^\circ - 29^\circ - 53^\circ = 98^\circ$$
d) Given angle $112^\circ$ and angle $d$ are complementary to $180^\circ$ (straight line).
$$d = 180^\circ - 112^\circ = 68^\circ$$
e) Angles $121^\circ$, $x$, and $y$ around a point.
Property: Angles around a point sum to $360^\circ$ and vertical angles equal.
Assuming $x$ and $y$ are vertical or supplementary, if $x$ and $y$ are supplementary to $121^\circ$:
$$x + y = 180^\circ - 121^\circ = 59^\circ$$
(More info needed for exact values.)
f) Given $18^\circ$ and $f$ form a right angle.
Property: Complementary angles sum to $90^\circ$.
$$f = 90^\circ - 18^\circ = 72^\circ$$
g) Angles $45^\circ$, $g$, and $24^\circ$ around a point on a straight line.
$$g = 180^\circ - 45^\circ - 24^\circ = 111^\circ$$
h) Angles $19^\circ$, $155^\circ$, and $h$ around intersecting lines.
$$h = 180^\circ - 155^\circ = 25^\circ$$
i) Two angles $x$ and $96^\circ$ on a straight line with $x$ repeated.
$$2x + 96^\circ = 180^\circ$$
$$2x = 180^\circ - 96^\circ = 84^\circ$$
$$x = \frac{84^\circ}{2} = 42^\circ$$
j) Angles $23^\circ$, $95^\circ$, and $x$ on a straight line.
$$x = 180^\circ - 23^\circ - 95^\circ = 62^\circ$$
2. **Larger angle is four times the smaller angle. Find both angles.**
Let smaller angle be $x$.
Larger angle is $4x$.
Angles form a straight line:
$$x + 4x = 180^\circ$$
$$5x = 180^\circ$$
$$x = \frac{180^\circ}{5} = 36^\circ$$
Larger angle:
$$4x = 4 \times 36^\circ = 144^\circ$$
3. **Find the size of $x$ in each:**
a) Angles $2x$ and $4x$ are supplementary.
$$2x + 4x = 180^\circ$$
$$6x = 180^\circ$$
$$x = 30^\circ$$
b) Angles $x$, $x$, and $2x$ sum to $180^\circ$.
$$x + x + 2x = 180^\circ$$
$$4x = 180^\circ$$
$$x = 45^\circ$$
c) Angles $5x$, $4x + 60$ sum to $180^\circ$.
$$5x + (4x + 60) = 180^\circ$$
$$9x + 60 = 180^\circ$$
$$9x = 120^\circ$$
$$x = \frac{120^\circ}{9} = 13.33^\circ$$
d) Angles $120^\circ$, $x$, and $2x$ sum to $180^\circ$.
$$120^\circ + x + 2x = 180^\circ$$
$$3x = 60^\circ$$
$$x = 20^\circ$$
e) Angles $78^\circ$, $3x$, $x$, and $2x$ sum to $360^\circ$ (around a point).
$$78^\circ + 3x + x + 2x = 360^\circ$$
$$78^\circ + 6x = 360^\circ$$
$$6x = 282^\circ$$
$$x = 47^\circ$$
f) Angles $x$ and $3x$ are supplementary.
$$x + 3x = 180^\circ$$
$$4x = 180^\circ$$
$$x = 45^\circ$$