Math Problem Statement

The diagram shows a quadrilateral PQRS and a straight line RT. PQ is parallel to RT. Given angles are: angle SPQ = 119 degrees, angle PQS = 81 degrees, and angle RST = 82 degrees. (a) Work out the size of angle x. (b)(i) Find the size of angle y. (ii) Give a reason for your answer.

Solution

Let's go through the problem step by step.

Given Information:

  1. The quadrilateral PQRSPQRS has a straight line RTRT.
  2. PQRTPQ \parallel RT.
  3. Angle SPQ=119\angle SPQ = 119^\circ.
  4. Angle PQS=81\angle PQS = 81^\circ.
  5. Angle RST=82\angle RST = 82^\circ.
  6. We need to find angles xx and yy.

Solution:

Part (a): Find the size of angle xx

Since PQRTPQ \parallel RT, we can use the concept of corresponding angles. In this configuration, angles PQS\angle PQS and RQT\angle RQT are corresponding angles. Hence, x=PQS=81x = \angle PQS = 81^\circ

So, x=81x = 81^\circ.

Part (b)(i): Find the size of angle yy

To find angle yy, we use the fact that the angles around point RR (angles xx, yy, and RST\angle RST) should add up to 180180^\circ because they form a straight line.

x+y+RST=180x + y + \angle RST = 180^\circ Substitute the known values: 81+y+82=18081^\circ + y + 82^\circ = 180^\circ 163+y=180163^\circ + y = 180^\circ y=180163y = 180^\circ - 163^\circ y=17y = 17^\circ

So, y=17y = 17^\circ.

Part (b)(ii): Give a reason for your answer

The reason for angle yy being 1717^\circ is that the angles around a straight line sum up to 180180^\circ. Therefore, we used this property to calculate yy based on the given angles xx and RST\angle RST.


Would you like further clarification on any part of the solution?

Related Questions:

  1. What is the significance of parallel lines in angle calculations?
  2. How do corresponding angles help in determining unknown angles in parallel line setups?
  3. Can you explain how angles around a straight line sum up to 180180^\circ?
  4. How would the solution change if PQPQ and RTRT were not parallel?
  5. What other properties of quadrilaterals can help in solving angle-related problems?

Tip:

Always look for parallel lines and corresponding angles in geometry problems involving quadrilaterals and transversals; they often simplify the solution process significantly.

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Math Problem Analysis

Mathematical Concepts

Plane Geometry
Angles
Parallel Lines
Quadrilaterals

Formulas

Sum of angles around a point = 180 degrees
Properties of corresponding angles in parallel lines

Theorems

Corresponding Angles Theorem
Angles around a straight line sum to 180 degrees

Suitable Grade Level

Grades 8-10