교육자를 위해: Angle rules and parallel lines(Mathematics, Geometry and measures)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트 — 수업에 사용하거나, 학습자들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.
Angles at a point sum to 360°: a + b + c + d = 360°.
Angles on a straight line sum to 180°: a + b + c = 180°.
Vertically opposite angles are equal: when two lines cross, the angles opposite each other are the same.
A right angle is 90°, often marked with a small square.
Complementary angles sum to 90°; supplementary angles sum to 180°.
Angles around a point
Parallel Lines and Transversals
Parallel lines are marked with matching arrows and never meet.
A transversal intersects two or more lines at distinct points; those lines need not be parallel. The equal-angle rules below require parallel lines.
Corresponding angles are equal: they occupy matching positions on the parallel lines and the transversal.
Alternate angles are equal: they lie on opposite sides of the transversal and between the parallel lines (Z shape).
Co-interior angles (also called allied or interior angles) sum to 180°: they lie between the parallel lines on the same side of the transversal (C shape).
Do not confuse alternate (equal) with co-interior (sum to 180°).
Alternate interior angles
Triangles
The interior angles of a triangle sum to 180°: a + b + c = 180°.
An equilateral triangle has all sides equal and all angles 60°.
An isosceles triangle has two equal sides and two equal base angles.
A right-angled triangle has one 90° angle; the other two angles sum to 90°.
The exterior angle of a triangle equals the sum of the two opposite interior angles.
Never assume a triangle is isosceles just from its appearance; check given markings or information.
Quadrilaterals
The interior angles of a quadrilateral sum to 360°.
A square has four equal sides and four right angles.
A rectangle has opposite sides equal and four right angles.
A parallelogram has opposite sides parallel and equal, and opposite angles equal.
A rhombus has all sides equal, opposite angles equal, and diagonals bisect at right angles.
A trapezium has one pair of parallel sides.
A kite has two pairs of adjacent equal sides and one pair of equal opposite angles.
Polygons: Interior Angles
For a simple polygon with n sides, the sum of interior angles is (n − 2) × 180°.
For a regular polygon (all sides and angles equal), each interior angle is \frac{(n-2) \times 180°}{n}.
A triangle (n=3) gives 180°; a quadrilateral (n=4) gives 360°; a pentagon (n=5) gives 540°.
To find the number of sides from an interior angle, rearrange the formula: n = \frac{360°}{180° - \text{interior angle}} (for regular polygons).
Be careful not to use the interior angle formula when you need the exterior angle.
Regular hexagon
Polygons: Exterior Angles
For a convex polygon, one exterior turn at each vertex sums to 360°. For concave simple polygons use signed turning angles.
For a regular polygon, each exterior angle is \frac{360°}{n}, where n is the number of sides.
To find the number of sides given an exterior angle e, use n = \frac{360°}{e}.
The interior and exterior angles at a vertex are supplementary: they sum to 180°.
If a regular polygon has an exterior angle of 40°, it has 9 sides.
Reasoning and Justification
Always give a reason for each step when finding angles, using correct terminology.
Common reasons: 'angles on a straight line sum to 180°', 'angles at a point sum to 360°', 'vertically opposite angles are equal', 'corresponding angles are equal', 'alternate angles are equal', 'co-interior angles sum to 180°', 'interior angles of a triangle sum to 180°', 'interior angles of a quadrilateral sum to 360°', 'exterior angles sum to 360°'.
State the angle fact you used, not just the calculation.
If a diagram shows parallel lines, you may use the parallel line properties; otherwise, do not assume they are parallel.
When proving two angles equal, identify the relevant angle relationship and write a clear chain of reasoning.
Common Mistakes to Avoid
Assuming lines are parallel without markings or given information.
Confusing alternate (equal) with co-interior (sum to 180°).
Using the interior angle formula when the exterior angle is needed.
Assuming a triangle is isosceles from appearance only.
Omitting reasons in angle calculations – you lose marks even if the final answer is correct.
Forgetting that exterior turns around a convex polygon sum to 360°.
Corresponding angles
Corresponding angles occupy the same relative position at the two intersections.
When the two lines are parallel, corresponding angles are equal.
Corresponding angles
Co-interior angles
Co-interior angles lie between the parallel lines on the same side of the transversal.
For parallel lines their sum is 180°. They are equal only in the special case of two right angles.
Co-interior angles
슬라이드
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4.Two angles are alternate angles on parallel lines. One is 70°. What is the other?
Easy
A70°
B110°
C180°
D90°
5.Two angles are co-interior angles on parallel lines. One is 70°. What is the other?
Easy
A110°
B70°
C180°
D90°
6.Which of the following statements are true for parallel lines cut by a transversal? (select all that apply) Select the relationships that hold in general, rather than only in a special right-angle case.
Medium
ACorresponding angles are equal.
BAlternate angles sum to 180°.
CCo-interior angles sum to 180°.
DVertically opposite angles are equal.
ECorresponding angles sum to 180°.
7.Two parallel lines are cut by a transversal. Match each angle description with its value.
Easy
Alternate interior angle to 70°
Corresponding angle to 50°
Co-interior angle to 40°
70°
50°
140°
8.Arrange the steps to find the sum of interior angles of a hexagon.
Medium
Use the formula: sum = (n - 2) × 180°
Substitute n = 6
Calculate (6 - 2) × 180° = 720°
더 많은 문제
9.Triangle ABC has angles A = 50°, B = 70° and C = 60°. Classify it by its side lengths.
Medium
AEquilateral
BIsosceles
CScalene
DRight-angled
10.Arrange the steps to find the size of angle x in a triangle where two angles are 50° and 60°.
Medium
Write the angle sum of a triangle: x + 50° + 60° = 180°
Simplify: x + 110° = 180°
Subtract 110° from both sides: x = 70°
11.Two parallel lines are cut by a transversal. Which of the following statements are true? (Select all that apply) Select the relationships that hold in general, rather than only in a special right-angle case.
Medium
ACorresponding angles are equal
BAlternate angles are equal
CCo-interior angles are equal
DCo-interior angles sum to 180°
12.Two angles are corresponding angles on parallel lines. One is 70°. What is the other?
Medium
A70°
B110°
C180°
D90°
13.A regular polygon has an interior angle of 150°. How many sides does it have?
Medium
A12
B10
C8
D15
14.two parallel lines are cut by a transversal. One corresponding angle is 72°. What is the size of the corresponding angle on the other parallel line?
Medium
A72°
B108°
C18°
D90°
15.If two parallel lines are cut by a transversal, then alternate angles are supplementary (sum to 180°).
Medium
True or false?
16.Two parallel lines are cut by a transversal. Which of the following pairs of angles are equal? (Select all that apply) Select the relationships that hold in general, rather than only in a special right-angle case.
Medium
AAlternate angles
BCorresponding angles
CCo-interior angles
DVertically opposite angles
17.A triangle has angles 2x, 3x, and 4x. What is the value of x?
Medium
A10
B20
C30
D40
18.A regular polygon has an interior angle of 144°. How many sides does it have?
Medium
A8
B9
C10
D12
19.Arrange the steps to find the number of sides of a regular polygon given that its exterior angle is 30°.
Medium
Write the formula: exterior angle = 360° / n
Substitute the given exterior angle: 30° = 360° / n
Solve for n: n = 360° / 30° = 12
20.A transversal cuts two parallel lines. Angle a is 65° and is corresponding to angle b. What is the size of angle b?
Medium
A65°
B115°
C25°
D180°
21.Which of the following are true for any triangle? (Select all that apply)
Medium
AInterior angles sum to 180°
BExterior angle equals sum of two opposite interior angles
CAll sides are equal
DIt has exactly one right angle
22.Alternate angles on parallel lines are equal.
Easy
True or false?
23.Arrange the steps to find the size of each interior angle of a regular polygon with 10 sides.
Medium
Calculate the sum of interior angles: (10-2)×180 = 1440°
Divide by the number of sides: 1440 ÷ 10 = 144°
State the answer: each interior angle is 144°
24.A regular polygon has 12 sides. What is the size of each exterior angle?
Medium
A30°
B150°
C12°
D45°
25.Which of the following are correct statements about angles? (Select all that apply) Select the relationships that hold in general, rather than only in a special right-angle case.
Medium
AVertically opposite angles are equal
BAngles on a straight line sum to 180°
CCo-interior angles on parallel lines are equal
DThe sum of interior angles of a quadrilateral is 360°
26.Co-interior angles on parallel lines are always equal.
Easy
True or false?
27.A regular polygon has an interior angle of 156°. How many sides does it have?
Medium
A15
B12
C10
D18
28.Match each angle property to its value.
Medium
Angles at a point
Angles on a straight line
Vertically opposite angles
Sum to 360°
Sum to 180°
Equal
29.Which of the following are correct for a parallelogram? (Select all that apply)
Medium
AOpposite angles are equal
BAdjacent angles sum to 180°
CAll sides are equal
DDiagonals bisect each other
30.A transversal cuts two parallel lines. One co-interior angle is 110°. What is the other co-interior angle?
Medium
A70°
B110°
C80°
D90°
31.Vertically opposite angles are equal.
Easy
True or false?
32.two parallel lines are cut by a transversal. One alternate angle is 3x + 10° and the other is 4x - 20°. Find the value of x.
Medium
A30
B20
C25
D15
33.A transversal cuts two parallel lines. One corresponding angle is 3x - 10° and the other is 2x + 20°. What is the value of x?
Medium
A30
B20
C10
D40
34.The sum of the interior angles of a quadrilateral is 360°.
Easy
True or false?
35.Match each polygon to the sum of its interior angles.
Medium
Triangle
Quadrilateral
Pentagon
180°
360°
540°
36.Arrange the steps to find the size of the largest angle in a triangle with angles in the ratio 2:3:4.
Medium
Let the angles be 2x, 3x, and 4x
Set up the equation 2x+3x+4x=180
Solve to get x=20
Largest angle = 4×20 = 80°
37.A regular polygon has an exterior angle of 15°. What is the sum of its interior angles?
Hard
A3960°
B3240°
C3600°
D2880°
38.Taking one exterior angle at each vertex of a convex polygon, in the same turning direction, gives a total of 360°.
Easy
True or false?
39.In a regular polygon, each exterior angle is 45°. How many sides does the polygon have?
Medium
40.Complete the sentence.
Easy
Angles on a straight line add up to ____ degrees.
41.In triangle ABC, A = 50° and B = 70°. Find C in degrees.
Easy
42.A regular polygon has an exterior angle of 40°. How many sides does it have?
Medium
43.Complete the sentence:
Easy
The sum of the interior angles of a quadrilateral is ____ degrees.
44.Complete the sentences about parallel lines:
Medium
Alternate angles are ____. Co-interior angles sum to ____ degrees.
45.two parallel lines are cut by a transversal. One angle is 120°. What is the size of the corresponding angle?
Medium
46.In a triangle, angle A = 2x, angle B = 3x, and angle C = 4x. Find the value of x.
Hard
47.Two parallel lines are cut by a transversal. One alternate angle is 65°. What is the size of the other alternate angle in degrees?
Easy
48.Two parallel lines are cut by a transversal. One co-interior angle is 115°. What is the size of the other co-interior angle in degrees?
Easy
49.In a triangle, angle A = 50°, angle B = 60°. What is the size of angle C in degrees?
Medium
50.Find the interior angle of a regular polygon with 8 sides. Give your answer in degrees.
Medium
51.two parallel lines are cut by a transversal. One angle is 120°. Explain why the co-interior angle is 60°, giving a reason.
Hard
52.two parallel lines are cut by a transversal. One angle is 60°. Explain why the corresponding angle is also 60°, giving a reason.
Medium
53.two parallel lines are cut by a transversal. One alternate angle is 50°. Explain why the other alternate angle is also 50°, giving a reason.
Hard
54.Explain why the sum of the interior angles of a pentagon is 540°, giving a reason.
Hard
55.two parallel lines are cut by a transversal. One co-interior angle is 120°. Explain why the other co-interior angle is 60°, giving a reason.
Hard
56.In a regular polygon, each exterior angle is 45°. How many sides does the polygon have?
Medium
A10
B8
C7
D9
57.In triangle ABC, A = 50° and B = 70°. Find C in degrees.
Easy
A66 °
B54 °
C72 °
D60 °
58.A regular polygon has an exterior angle of 40°. How many sides does it have?
Medium
A10
B11
C8
D9
59.In a triangle, angle A = 2x, angle B = 3x, and angle C = 4x. Find the value of x.
Hard
A20 °
B22 °
C24 °
D18 °
60.Two parallel lines are cut by a transversal. One alternate angle is 65°. What is the size of the other alternate angle in degrees?
Easy
A72 °
B65 °
C58 °
D79 °
61.Two parallel lines are cut by a transversal. One co-interior angle is 115°. What is the size of the other co-interior angle in degrees?
Easy
A58 °
B79 °
C65 °
D72 °
62.In a triangle, angle A = 50°, angle B = 60°. What is the size of angle C in degrees?
Medium
A70 °
B63 °
C77 °
D84 °
63.Find the interior angle of a regular polygon with 8 sides. Give your answer in degrees.
Medium
A135 °
B149 °
C163 °
D121 °
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