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mathematics UK Year 10 Mathematics (KS4)
 
Solved problems for the chapter:
 

Lines and angles

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Detailed solution steps are available at:
 
https://en.openprof.com/wb/chapter:lines_and_angles/5337/


Example 1.

The three vertices of a quadrilateral are given ABCD. , , , the vertex C lies on the abscissa.


  • Draw the quadrilateral.


  • Determine the coordinate of the point C if the area of the quadrilateral ABCD is equal to .


  • Calculate the length of the diagonal AC.

    
Example 2.

Express the given angle in the form of degrees, minutes and seconds:


    
3.

The coordinates of the points , and are given in the plane. Calculate height to the side in the triangle .

    
4.

Converts the given angle to degrees:


    
5.

The following angle is given:




Subtract the half of the complementary angle of from its complementary angle.


Name the new angle, .

    
6.

The diagram below shows an isosceles triangle ABC. If in an isosceles triangle is 28°, and the height at base c is 6 cm. Calculate the angle and the side of a triangle .


    
7.

Given that the angles and are supplementary, and that the angles and are complementary. If is three times larger than the angle , calculate the values of , and .

    
8.

The diagram below shows an isosceles triangle ABC. The area of the isosceles triangle is 15.49 , the angle is 29°, and the perimeter is 20 cm. Calculate the sides of the isosceles triangle.


    

9.

The difference between three times the angle and two times the angle represents a right angle, and the sum of two times the value of the angle and the three times the value of the angle is supplementary. Find the measure of the angles and to the nearest angular second.

    
10.

The diagram below shows an isosceles triangle ABC. If in the isosceles triangle is 76°, and side a is 8.24 cm. Calculate , the base and the height to the base of the triangle .


    
11.

The angle measures 45% of the sum of angles on a straight line. How many times larger is to its complementary angle? Also, calculate the complementary and supplementary of the angle .

    
12.

Given two supplementary angles and in the ratio 5:7, find the measures of these angles.

    
13.


Given the diagram above with , determine the value of angles , and .

    
14.

The sketch made bellow is , and is the angle bisector . The size of the angle is 30°. Calculate .


    
15.

At what angle do the bisectors of the exterior angles intersect at the hypotenuse of a right-angled triangle?


Given data:






    
16.

The figure below shows a square and two equilateral triangles:




First construct the line with the ends E and F, and then draw the line holders DE and BF. Mark the intersection of the girders with point G.


Determine the interior angles of triangle EFG! What type of triangle is it?

    
17.

Calculate the minimum height in a triangle with sides and .

    
18.

The figure below shows two squares and two equilateral triangles:




Construct a line with vertices B and D and determine the interior angles of triangle BDC! What triangle is it?

    

19.

Th length of the longer diagonal of a kite is 11 cm and one of its interior angle is 90.41 °. The perimeter of the kite measures 28 cm and the area 46.84 . Calculate the length of the longer side and the remaining interior angles of the kite.

    
20.

At what angle does the angle bisector and the altitude h on side a intersect?


Given data:






    
21.

The two sides of a kite are given as 5 and 3 cm with the shorter diagonal as 3.71 cm. Calculate the interior angles of the kite.

    
22.

The ratio of the sides of a kite is 8: 6 and the interior angle between the smaller sides measures 67 °. Calculate all remaining interior angles of the kite.

    
23.

The figure below shows two squares and an isosceles triangle. Angle is 63.5°.




First construct a line with ends E and G, and then draw the line holders AB and CD. Mark the intersection of the beams with point H.


Determine the interior angles of the pentagon AHDEG!

    
24.

The diagram below shows a right-angled triangle ABC. The area of the triangle is 21.65 . The length of the hypotenuse c is 10 and the perimeter is 23.66 . Calculate the length of the sides.


    
25.

The figure below shows an isosceles triangle, with angle = 40°.




First draw the bisectors of the interior angles along the base, and then construct a parallel to the base through vertex C. Mark the resulting intersections with points D, E, and F. Points D, E, and F are the vertices of the geometric figure. Determine the interior angles of the triangle DEF.

    
26.

Given an isosceles triangle ABC with angle and it's perimeter 15cm. Calculate the angle of the bisector of the sides of the triangle and the center of gravity on the enclosed side.


    
27.

In an acute-angled triangle ABC, the angle between the height and the side b is , and the angle is . Make a sketch and calculate the remaining interior angles of the triangle ABC.

    
28.

In an isosceles triangle ABC, the base is 4.2 cm and the sides are 6.5 cm each. Make a sketch of the isosceles triangle, mark the angle between the base and the sides and calculate its size to the nearest degree.

    

29.

An interior angle of a trapezium measures 46 °. Calculate the remaining angles of the trapezium.

    
30.

The midline of a rectangular trapezium measures 8.07 cm, height 4 cm, and perimeter 25.90 cm. Calculate all interior angles of the trapezium.

    
31.

Two parallel sides of a rectangular trapezium measure 8.14 cm and 4 cm, the height is 4.47 cm. Calculate the diagonals of the trapezium.

    
32.

The two legs of a rectangular trapezium are in the ratio 8: 11. Calculate all interior angles in the trapezium.

    
33.

The longer leg of a rectangular trapezium measures 7 cm, its height is 6.32 cm and perimeter is 22.32 cm. Calculate the remaining sides of the trapezium.

    
34.

The parallel sides of a trapezium measure 9cm and 11cm, while its height is 6cm. If a side and the height are equal and two of its interior angles are right angle, calculate the length of the remaining side and its acute interior angle to 1 decimal place.

    
35.

The height of a rhombus measures 9 cm. If its acute angle is . Calculate the length of its side and longer diagonal of the rhombus.

    
36.

A rhombus has a shorter diagonal of length and its interior acute angle as . Calculate the length of its side.

    
37.

One of the interior angles of a trapezium measures . If the slant sides are equal, calculate the other interior angle.

    
38.

The area of a right-angled triangle is 12.25 and the angle is 44.42°. Calculate the length of the hypotenusec and the perimeter of the triangle.

    

39.

Side a and b in a right-angled triangle is 3 and 8 cm respectively. Calculate the length of the hypotenuse and the interior angles in the triangle.

    
40.

Side b in a right-angled triangle is 8 cm and the hypotenuse is 10.63 cm. Calculate the length of side a and the interior angles in the triangle.

    
41.

The angle in a right-angled triangle is 27°, and the length of side a is 3 cm. Calculate the remaining sides and interior angles in the triangle.

    
42.

The angle in a right-angled triangle is 69° and the length of the hypotenuse is 8.37 cm. Calculate the sides and the remaining interior angles in the triangle.

    
43.

Given a vertical wall 4.8 m high. Matthew has a 5.2 m long ladder. At what angle to the flat floor must he lay it to reach exactly the top of the wall? Make a sketch.

    
44.

Calculate the length of the hypotenuse of the right-angled triangle in the sketch below:


    
45.

The sketch below shows a right-angled triangle. Calculate the sides of the triangle.


    
46.

The triangle below has a circumference of 36 cm. Calculate the length of the sides and the area of the triangle


    
47.

In triangle ABC, the interior angle at vertex A is 53° and the interior angle at vertex B is 71°.


Make a sketch of triangle ABC. Calculate the interior angle at vertex C, and the outer angle at vertex B, .

    
48.

The interior angle of a right-angled triangle is . Calculate the remaining interior angles and the corresponding exterior angles of this triangle. Enter the results in the table below:


Inner angles of a triangleExternal angles of a triangle 
 
 
 
    

49.

The sketch below shows a right-angled triangle.




Calculate to the nearest minute, the angle and .

    
50.

The sketch below shows an isosceles triangle ABC (| AC | = | BC |). Calculate the interior angles of the triangle.


    
51.

The outer angle is three times larger than the outer angle of a triangle, and the inner angle is 100 ° smaller than the inner angle . Calculate the interior and exterior angles of the triangle.

    
52.

The exterior angles of a triangle are in the ratio




Determine all interior and exterior angles of the triangle.

    
53.

In a triangle, the angle is three times the angle , and the angle is half the angle . Calculate its interior

angles and state the type of triangle it is.

    
54.

Right-right triangle ABE has sides 8, 15 and 17 cm long. Calculate the lenghts of diagonals e and f of the rhombus ABCD.


    
55.

In the figure drawn below, the side AB is of the same length as the side AD. At what angle do the diagonals e and f intersect?


    
56.

The angle between the diagonals of the in the rhombus below is 35 degrees greater than the angle . Calculate the interior angle .


    
57.

The angles and the exterior angle are equal. Calculate the sizes of the interior angles of the rhombus


    
58.

What is the length c of the ladder that Peter needs to install gutters on the house if ? Express the result in meters.


    

59.

Calculate the interior and exterior angles of a triangle in which angle is 3.5 times greater than angle and angle is 1.5 times greater than angle .


Interior and exterior angle markings

    
60.

The angle of a triangle is equal to the sum of angle and three halves of angle . Calculate the angles of a triangle if angle is equal to four fifths of angle where are the interior angles of the triangle.