Question

section 5.3 homework
page 278: 1-9, 20
name:
date:
period:
use your conjectures to find the missing measures.

1. abcd is a kite.

perimeter =

1. x =

y =

1. x =

y =

1. x =

perimeter = 85 cm

Explanation:

Problem 1: Kite Perimeter

Step1: Recall Kite Properties

A kite has two pairs of adjacent sides equal. So \( AB = AD = 20 \, \text{cm} \) and \( BC = CD = 12 \, \text{cm} \).

Step2: Calculate Perimeter

Perimeter is sum of all sides: \( 20 + 20 + 12 + 12 \).
\( 20 + 20 = 40 \), \( 12 + 12 = 24 \), \( 40 + 24 = 64 \).

Step1: Sum of Quadrilateral Angles

Sum of interior angles of a quadrilateral is \( 360^\circ \). Given one angle \( 146^\circ \), another \( 47^\circ \), and assume the quadrilateral is a kite (two pairs of adjacent angles? Wait, the diagram shows two pairs of equal sides. Wait, the angles: let's see, in a quadrilateral, sum is \( 360^\circ \). Let's assume the two angles marked with \( x \) are equal? Wait, no, the diagram: one angle is \( 146^\circ \), another is \( 47^\circ \), and two angles \( x \) and \( y \)? Wait, maybe it's a kite? Wait, the user's work shows \( 180(4 - 2) = 360 \). So \( x + x + 146 + y = 360 \)? Wait, no, maybe the quadrilateral has two pairs of equal angles? Wait, the given angle is \( 146^\circ \), and \( 47^\circ \), and two angles \( x \) and \( y \). Wait, maybe it's a parallelogram? No, the diagram looks like a kite. Wait, the user's calculation: \( x = 360 - 146 - 146 - 47 \)? Wait, no, let's recalculate. Sum of angles: \( 360^\circ \). If one angle is \( 146^\circ \), another is \( 47^\circ \), and maybe two angles: let's see, in a kite, one pair of opposite angles? No, kite has one pair of opposite angles equal? Wait, no, kite has two pairs of adjacent sides equal. So the angles between the unequal sides are equal. Wait, maybe the quadrilateral has angles: \( 146^\circ \), \( 47^\circ \), \( x \), \( y \), and \( x = 47^\circ \)? No, the user's work: \( x = 360 - 146 - 146 - 47 \)? Wait, no, let's do it properly. Let's say the quadrilateral has angles \( 146^\circ \), \( y \), \( 47^\circ \), \( x \). If it's a kite with two pairs of adjacent sides equal, then the angles between the equal sides: so maybe \( x = 47^\circ \)? No, that doesn't make sense. Wait, the sum is \( 360 \). So \( x + y + 146 + 47 = 360 \). If the quadrilateral is a parallelogram? No, the diagram has two pairs of equal sides (marked with ticks). So it's a kite? Wait, no, two pairs of equal sides make a parallelogram? No, two pairs of adjacent sides equal make a kite. So in a kite, one pair of opposite angles are equal. Wait, maybe \( y = 146^\circ \), and \( x = 47^\circ \)? No, that would sum to \( 47 + 146 + 47 + 146 = 386 \), which is more than 360. So that's wrong. Wait, the user's work: \( 180(4 - 2) = 360 \), then \( x = 360 - 146 - 146 - 47 \)? Wait, no, \( 360 - 146 - 47 - y = x \). Wait, maybe the angle \( y \) is equal to \( 146^\circ \), so \( x = 360 - 146 - 146 - 47 = 360 - 339 = 21^\circ \), and \( y = 146^\circ \). Let's check: \( 21 + 146 + 47 + 146 = 21 + 47 = 68, 146 + 146 = 292, 68 + 292 = 360 \). Yes! So \( x = 21^\circ \), \( y = 146^\circ \).

Step1: Figure Type

The figure has two pairs of adjacent sides equal? Wait, the marks: two sides on the left are equal, two on the right? Wait, it's a quadrilateral with three sides marked equal? No, the marks: left side has a tick, bottom side has a tick, right side has a tick, top side? Wait, no, the diagram: left side (vertical) has a tick, bottom side has a tick, right side (vertical) has a tick, top side? Wait, maybe it's a rhombus? No, one angle is \( 128^\circ \). Wait, in a quadrilateral with two pairs of adjacent sides equal (a kite? No, three sides? Wait, the marks: left side (vertical) has one tick, bottom side has one tick, right side (vertical) has one tick, top side? Wait, maybe it's an isosceles trapezoid? No, the angle is \( 128^\circ \). Wait, in a quadrilateral with two pairs of equal sides (a parallelogram), consecutive angles are supplementary. But here, one angle is \( 128^\circ \), so the adjacent angle would be \( 180 - 128 = 52^\circ \). Wait, the marks: left side and right side have ticks (equal), bottom and top? No, the left side (vertical) has a tick, bottom side has a tick, right side (vertical) has a tick, top side? Wait, maybe it's a kite with two pairs of adjacent sides equal: left and top, bottom and right? No, the angle is \( 128^\circ \). Wait, the sum of interior angles: \( 360^\circ \). If it's a parallelogram, opposite angles are equal, consecutive supplementary. But here, one angle is \( 128^\circ \), so the opposite angle is \( 128^\circ \), and the other two are \( 52^\circ \). Wait, the marks: left side (vertical) has a tick, right side (vertical) has a tick (so they are equal), bottom side has a tick, top side? Wait, maybe \( x = 52^\circ \), \( y = 128^\circ \)? No, that doesn't fit. Wait, the user's diagram: a quadrilateral with left side (vertical) having one tick, bottom side having one tick, right side (vertical) having one tick, top side? Wait, maybe it's a rhombus? No, angle is \( 128^\circ \). Wait, let's think again. The quadrilateral has three sides with ticks? No, two sides: left (vertical) and right (vertical) have ticks (equal), bottom and top? No, bottom has a tick, top? Wait, maybe it's an isosceles trapezoid with legs equal (left and right sides) and bases? No, the angle is \( 128^\circ \). In an isosceles trapezoid, base angles are equal, and consecutive angles are supplementary. So if one angle is \( 128^\circ \), the adjacent angle is \( 52^\circ \). So \( x = 52^\circ \), \( y = 128^\circ \)? Wait, the marks: left side (vertical) and right side (vertical) have ticks (equal, legs), bottom and top? No, bottom has a tick, top? Wait, maybe \( x = 52^\circ \), \( y = 128^\circ \).

Answer:

\( 64 \, \text{cm} \)

Problem 2: Quadrilateral Angles