Basic Angle Properties

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수업 노트

Angles around Points & Lines

Angles around a point add up to 360°.
Angles on a straight line add up to 180°.
Vertically opposite angles are equal when two lines intersect.
Use known angle properties to find missing angles, even if they are not the target angle.

Vertically opposite angles

Angles in Triangles

The three interior angles of any triangle sum to 180°.
In an isosceles triangle, the two angles opposite the equal sides are equal.
In an equilateral triangle, all three angles are 60°.
A right-angled triangle has one 90° angle.

Triangle types

Angles in Quadrilaterals

The four interior angles of any quadrilateral sum to 360°.
In a square or rectangle, all angles are 90°.
In a parallelogram or rhombus, opposite angles are equal.
In a kite, one pair of opposite angles are equal.

Quadrilateral types

Problem-Solving Tips

Mark all missing angles you can find on the diagram using angle properties.
Use vertically opposite angles and straight line angles to find triangle angles.
For isosceles triangles, identify the equal angles from the equal sides.
Combine triangle and quadrilateral angle sums with straight line facts.

슬라이드

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연습 문제

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1.What is the sum of angles around a point?
Easy
- A180°
- B360°
- C90°
- D270°
2.Angles that form a straight line add up to
Easy
- A90°
- B180°
- C360°
- D270°
3.Vertically opposite angles are
Easy
- Aequal
- Bsupplementary
- Ccomplementary
- Ddifferent
4.The three interior angles of any triangle add up to
Easy
- A90°
- B180°
- C360°
- D270°
5.In an equilateral triangle, each interior angle is
Easy
- A45°
- B60°
- C90°
- D30°
6.A right-angled triangle has one angle equal to
Easy
- A45°
- B60°
- C90°
- D180°
7.The sum of interior angles of a quadrilateral is
Easy
- A180°
- B270°
- C360°
- D90°
8.In a rectangle, each interior angle is
Easy
- A45°
- B60°
- C90°
- D180°

더 많은 문제

9.Two angles on a straight line measure 70° and x. Find x.
Easy
- A110°
- B70°
- C20°
- D180°
10.In triangle ABC, angle A = 50°, angle B = 60°. Find angle C.
Easy
- A70°
- B60°
- C50°
- D80°
11.An isosceles triangle has a vertex angle of 40°. Find the base angles.
Medium
- A70° each
- B40° each
- C50° each
- D60° each
12.In a quadrilateral, three angles are 80°, 100°, and 120°. Find the fourth angle.
Medium
- A60°
- B70°
- C50°
- D80°
13.Two intersecting lines create an angle of 35°. Find the vertically opposite angle.
Easy
- A35°
- B145°
- C55°
- D70°
14.Triangle ABC is isosceles with AB = AC. Angle ABC = 32°. Find angle BAC.
Medium
- A116°
- B148°
- C64°
- D32°
15.In a triangle, angles are x, 2x, and 3x. Find x.
Medium
- A30°
- B60°
- C90°
- D45°
16.A student measures triangle angles as 55°, 85°, and 50°. Why is this impossible?
Medium
- ASum is 190°, not 180°
- BSum is 180°, but angles are too small
- COne angle is not acute
- DThe triangle cannot be drawn
17.In the diagram, ABC is a straight line. Angle ABD = 125°. Find angle DBC.
Medium
- A55°
- B65°
- C45°
- D35°
18.In triangle ABC, angle A = 40°, angle B = 70°. Find angle C.
Easy
- A70°
- B60°
- C50°
- D80°
19.A quadrilateral has angles 110°, 80°, 90°. Find the fourth angle.
Medium
- A80°
- B90°
- C100°
- D70°
20.In an isosceles triangle, the vertex angle is 50°. Find the base angles.
Medium
- A65° each
- B60° each
- C70° each
- D55° each
21.Two angles are vertically opposite. One is 73°. Find the other.
Easy
- A73°
- B107°
- C17°
- D90°
22.In triangle PQR, angle P = 2x, angle Q = 3x, angle R = 5x. Find x.
Medium
- A18°
- B20°
- C15°
- D10°
23.A triangle has angles in the ratio 2:3:4. Find the largest angle.
Medium
- A80°
- B60°
- C40°
- D100°
24.A regular polygon has an interior angle of 144°. How many sides does it have?
Medium
- A10
- B12
- C8
- D9
25.In a quadrilateral, angles are x, 2x, 3x, and 4x. Find x.
Medium
- A36°
- B30°
- C40°
- D45°
26.In the diagram, two lines intersect. One of the angles is 55°. What is the size of the vertically opposite angle?
Easy
- A35°
- B55°
- C125°
- D180°
27.A triangle has angles 45°, 75°, and x. What is x?
Easy
- A50°
- B60°
- C70°
- D80°
28.In a quadrilateral, three angles are 90°, 90°, and 80°. What is the fourth angle?
Easy
- A80°
- B90°
- C100°
- D110°
29.An isosceles triangle has a base angle of 40°. What is the vertex angle?
Medium
- A80°
- B100°
- C120°
- D140°
30.In triangle ABC, AB = AC. Angle BAC = 38°. BCD is a straight line. Find angle ACD.
Hard
- A71°
- B99°
- C109°
- D142°
31.A triangle has angles in the ratio 2:3:5. What is the largest angle?
Medium
- A54°
- B72°
- C90°
- D108°
32.In a parallelogram, one angle is 65°. What is the adjacent angle?
Easy
- A65°
- B115°
- C125°
- D180°
33.In triangle PQR, angle P = 2x, angle Q = 3x, angle R = 4x. Find x.
Medium
- A15°
- B20°
- C25°
- D30°
34.In the diagram, triangle BCD is isosceles with BC = BD. ABC is a straight line. Angle CBD = 40°. Find angle BDC.
Medium
- A40°
- B60°
- C70°
- D80°
35.Vertically opposite angles are always equal.
Easy
True or false?
36.In triangle ABC, angle A = 45°, angle B = 55°. What is angle C?
Easy
- A80°
- B100°
- C90°
- D70°
37.In a rectangle, all interior angles are 90°.
Easy
True or false?
38.Two intersecting lines form an angle of 42°. What is the size of the vertically opposite angle?
Easy
- A42°
- B138°
- C48°
- D90°
39.Match each angle property to its correct sum or description.
Medium
- Angles around a point
- Angles on a straight line
- Interior angles of a triangle
- Interior angles of a quadrilateral
- 180°
- 360°
- Add up to 180°
- Add up to 360°
40.Arrange the following steps in the correct order to find the missing angle in a triangle when two angles are given: 65° and 45°.
Medium
- Write the sum: 65° + 45° + x = 180°
- Subtract the sum from 180°: x = 180° - 110°
- Add the known angles: 65° + 45° = 110°
- State the answer: x = 70°
41.Which of the following statements about the angles in a kite are always true? (Select all that apply)
Medium
- AOne pair of opposite angles are equal
- BAll four angles are equal
- CAdjacent angles are supplementary
- DThe sum of interior angles is 360°
- EAll angles are 90°
42.Vertically opposite angles are always supplementary.
Easy
True or false?
43.In the diagram, two lines intersect. One angle is 3x and the vertically opposite angle is x + 40°. Find x.
Medium
- A10°
- B20°
- C30°
- D40°
44.In an isosceles triangle, the vertex angle is 70°. What is the size of each base angle?
Medium
- A55°
- B65°
- C70°
- D110°
45.Two angles on a straight line are 30° and x. Find x.
Easy
46.An isosceles triangle has a vertex angle of 30°. Find the size of each base angle.
Medium
47.Complete the sentence.
Easy
The sum of the interior angles of a quadrilateral is ____°.
48.In a quadrilateral, three angles are 70°, 110°, and 80°. Find the fourth angle.
Easy
49.In triangle ABC, AB = AC. Angle ABC = 35°. Find angle BAC.
Medium
50.In the diagram, ABC is a straight line. Angle ABD = 130°. Find angle DBC.
Easy
51.A triangle has angles in the ratio 2:3:5. Find the largest angle.
Medium
52.In a right-angled triangle, one acute angle is 37°. Find the other acute angle.
Easy
53.In a quadrilateral ABCD, angle A = 65°, angle B = 110°, angle C = 85°. Find angle D and explain your method.
Medium
54.Two angles on a straight line are 30° and x. Find x.
Easy
- A135 °
- B165 °
- C150 °
- D180 °
55.An isosceles triangle has a vertex angle of 30°. Find the size of each base angle.
Medium
- A83 °
- B67 °
- C91 °
- D75 °
56.In a quadrilateral, three angles are 70°, 110°, and 80°. Find the fourth angle.
Easy
- A110 °
- B100 °
- C90 °
- D120 °
57.In triangle ABC, AB = AC. Angle ABC = 35°. Find angle BAC.
Medium
- A132 °
- B121 °
- C110 °
- D99 °
58.A triangle has angles in the ratio 2:3:5. Find the largest angle.
Medium
- A99 °
- B81 °
- C108 °
- D90 °
59.In a right-angled triangle, one acute angle is 37°. Find the other acute angle.
Easy
- A58 °
- B63 °
- C48 °
- D53 °

복습 노트

## Basic Angle Properties ### Angles around Points & Lines - **Angles around a point** add up to **360°**. - **Angles on a straight line** add up to **180°**. - **Vertically opposite angles** are equal (when two lines intersect). ### Angles in Triangles - **Interior angles of a triangle** add up to **180°**. - **Isosceles triangle**: two sides equal → two base angles equal. - **Equilateral triangle**: all angles = 60°. - **Right-angled triangle**: one angle = 90°. ### Angles in Quadrilaterals - **Interior angles of a quadrilateral** add up to **360°**. - Square/rectangle: all angles = 90°. - Parallelogram/rhombus: opposite angles equal. - Kite: one pair of opposite angles equal. ### Tips - Mark all missing angles you can find on the diagram. - Use known angles to find others step by step. ### Example (from source) Find x and y where three lines intersect: vertically opposite gives x = 25°; angles on a straight line: 25 + y + 98 = 180 → y = 57°. ### Common Question Types - Find missing angle in triangle/quadrilateral. - Use isosceles property (base angles equal). - Combine straight line and triangle properties. - Explain why a triangle sum is wrong (e.g., 55+85+50=190 ≠ 180).

Angles around a point: Angles that meet at a point sum to 360°.
Angles on a straight line: Angles that form a straight line sum to 180°.
Vertically opposite angles: When two lines intersect, the angles opposite each other are equal.
Interior angles of a triangle: The three angles inside any triangle sum to 180°.
Isosceles triangle: A triangle with two equal sides; the angles opposite those sides are equal.
Equilateral triangle: A triangle with all sides equal; all angles are 60°.
Right-angled triangle: A triangle with one angle of 90°.
Interior angles of a quadrilateral: The four angles inside any quadrilateral sum to 360°.

플래시카드

What do angles around a point add up to?

Angles around a point add up to 360°.

What do angles on a straight line add up to?

Angles on a straight line add up to 180°.

What are vertically opposite angles?

Vertically opposite angles are equal. They occur when two lines intersect.

What do the interior angles of a triangle add up to?

The interior angles of a triangle add up to 180°.

In an isosceles triangle, which angles are equal?

The two angles opposite the equal sides are equal.

What are the angles in an equilateral triangle?

Each angle in an equilateral triangle is 60°.

What is the sum of interior angles in a quadrilateral?

The sum of interior angles in a quadrilateral is 360°.

In a parallelogram, what is true about opposite angles?

Opposite angles in a parallelogram are equal.

In a kite, which angles are equal?

One pair of opposite angles are equal in a kite.

What is the angle in a right-angled triangle that is 90°?

One angle in a right-angled triangle is 90°.

If a triangle has angles 55°, 85°, and 50°, why is it incorrect?

The sum is 190°, but the interior angles of a triangle must add up to 180°.

In an isosceles triangle, one angle is 96°. Find the other two angles.

The other two angles are 42° and 42° (since 180 - 96 = 84, divided by 2).

Kim says if one angle of an isosceles triangle is 48°, another must be 66°. Why is he wrong?

The 48° could be the vertex angle, making base angles (180-48)/2 = 66°, or it could be a base angle, making the other base angle also 48° and vertex 84°.

In triangle ABC, AB = AC and angle BAC = 38°. BCD is a straight line. Find angle ACD.

Angle ABC = (180-38)/2 = 71°. Angle ACD = 180 - 71 = 109° (angles on a straight line).

In triangle BCD, ABC is a straight line, BC = BD and angle CBD = 40°. Find angle BDC.

70°. Triangle BCD is isosceles (BC = BD), so the base angles are equal: (180° − 40°) ÷ 2 = 70°.

What is the value of x in a trapezium where three angles are given?

Sum of angles in a quadrilateral is 360°, so x = 360 - (sum of given angles).

In triangle PRS, Q is on PR, PS = QS, angle QRS = 25°, angle RSQ = 32°. Find angle PQS.

In triangle QRS, angle RQS = 180° − (25° + 32°) = 123°. Since P, Q, R lie on a straight line, angle PQS = 180° − 123° = 57°.

In triangle PRS, Q is on PR, PS = QS, angle QRS = 25°, angle RSQ = 32°. Find angle PSR.

Angle PQS = 57° (see the previous card). Triangle PQS is isosceles (PS = QS), so angle SPQ = angle SQP = 57°, giving angle PSQ = 180° − 57° − 57° = 66°. Angle PSR = angle PSQ + angle QSR = 66° + 32° = 98°.

What is the sum of angles in any quadrilateral?

360°.

If a quadrilateral is a square or rectangle, what are its angles?

All angles are 90°.

What is the value of x if angles around a point are 25°, 98°, and x?

x = 360 - (25+98) = 237°.

What is the value of y if angles on a straight line are 25°, y, and 98°?

y = 180 - (25+98) = 57°.

In an irregular quadrilateral, three angles are 97°, 115°, and 85°. Find the fourth angle.

Fourth angle = 360 - (97+115+85) = 63°.

If a triangle has angles x, 60°, and 50°, find x.

x = 180 - (60+50) = 70°.

What is the relationship between angles in a rhombus?

Opposite angles are equal.

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