Math / Geometry

Question2. Find the values of the unknowns $x$ and $y$ in the following-diagrams: (i) (ii) (iii)

Studdy Solution

STEP 1

1. The sum of the interior angles of a triangle is always $180^\circ$ .
2. An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

STEP 2

1. Solve for $x$ in diagram (i) using the sum of interior angles.
2. Solve for $x$ and $y$ in diagram (ii) using the sum of interior angles and the property of the exterior angle.
3. Solve for $x$ and $y$ in diagram (iii) using the sum of interior angles and the property of the exterior angle.

STEP 3

Use the sum of interior angles in a triangle to solve for $x$ in diagram (i).
$50^\circ + 120^\circ + x = 180^\circ$

STEP 4

Rearrange the equation to solve for $x$ .
$x = 180^\circ - 50^\circ - 120^\circ$

STEP 5

Calculate the value of $x$ .
$x = 10^\circ$

STEP 6

Use the sum of interior angles in a triangle to form an equation involving $x$ and $y$ in diagram (ii).
$50^\circ + x + y = 180^\circ$

STEP 7

Use the property of the exterior angle to form another equation for $x$ and $y$ in diagram (ii).
$80^\circ = 50^\circ + y$

STEP 8

Rearrange the second equation to solve for $y$ .
$y = 80^\circ - 50^\circ$

STEP 9

Calculate the value of $y$ .
$y = 30^\circ$

STEP 10

Substitute $y = 30^\circ$ into the first equation to solve for $x$ .
$50^\circ + x + 30^\circ = 180^\circ$

STEP 11

Rearrange the equation to solve for $x$ .
$x = 180^\circ - 50^\circ - 30^\circ$

STEP 12

Calculate the value of $x$ .
$x = 100^\circ$

STEP 13

Use the sum of interior angles in a triangle to form an equation involving $x$ and $y$ in diagram (iii).
$50^\circ + 60^\circ + y = 180^\circ$

STEP 14

Rearrange the equation to solve for $y$ .
$y = 180^\circ - 50^\circ - 60^\circ$

STEP 15

Calculate the value of $y$ .
$y = 70^\circ$

STEP 16

Use the property of the exterior angle in diagram (iii).
$x = 50^\circ + 60^\circ$

STEP 17

Calculate the value of $x$ .
$x = 110^\circ$
Solution: Diagram (i): $x = 10^\circ$ \\ Diagram (ii): $x = 100^\circ$ , $y = 30^\circ$ \\ Diagram (iii): $x = 110^\circ$ , $y = 70^\circ$

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