1 find the length of a, b, and c and the measure of ∠x, ∠y, and ∠z in the parallelograms below.
(1)
(2)
2 the following statements (1) to (4) represent the conditions for quadrilateral abcd to be called a parallelogram. add the missing part of each condition. question (3), and (4) will have two answers.
(1) ∠a = ∠c
(2) given the intersection point of ac and bd is o,
ao = co
(3) ab = cd
(4) ad ∥ bc
3 name the quadrilateral that fits to the following descriptions.
(1) a parallelogram where the diagonals are equal in length.
(2) a parallelogram where the adjacent sides are equal.

Question

Accepted Answer

### Problem 1 (1)
# Explanation:
## Step 1: Recall properties of parallelograms
In a parallelogram, opposite sides are equal. So $ AD = BC $ and $ AB = CD $, and opposite angles are equal, also consecutive angles in a triangle or related angles can be found using angle sum property.
Given $ BC = 6 $ cm, so $ a = AD = BC = 6 $ cm.
Given $ AB = 5 $ cm? Wait, no, in triangle $ ABC $, $ AB $ is a side, but in parallelogram $ ABCD $, $ AB = CD $, and $ BC = AD $. Also, in triangle $ BCD $ or $ ABC $, let's find $ \angle x $. In triangle $ ABC $, angles are $ 65^\circ $, $ 50^\circ $, so the third angle at $ A $ in triangle? Wait, no, in parallelogram $ ABCD $, $ AB \parallel CD $, so alternate angles? Wait, the triangle at $ C $: angle at $ B $ is $ 65^\circ $, angle at $ C $ in triangle $ ABC $ is $ 50^\circ $? Wait, maybe better to use parallelogram properties: opposite sides equal, so $ a = BC = 6 $ cm (since $ AD \parallel BC $ and $ AD = BC $ in parallelogram). Then $ b = AB = 5 $ cm? Wait, no, $ AB $ is 5 cm? Wait, the diagram shows $ BC = 6 $ cm, $ AB $ side? Wait, maybe $ AB = 5 $ cm? Wait, the triangle has sides 5 cm and 6 cm. Wait, in parallelogram $ ABCD $, $ AD = BC = 6 $ cm (so $ a = 6 $ cm), $ AB = CD = 5 $ cm? No, wait $ BC = 6 $ cm, so $ AD = BC = 6 $ cm (so $ a = 6 $ cm). Then $ b = AB = 5 $ cm? Wait, maybe. Then for $ \angle x $: in triangle $ BCD $, or using angle sum. Wait, in parallelogram, adjacent angles are supplementary? No, wait in triangle $ ABC $, angles at $ B $ is $ 65^\circ $, at $ C $ is $ 50^\circ $, so angle at $ A $ in triangle $ ABC $ is $ 180 - 65 - 50 = 65^\circ $. Then in parallelogram, $ \angle x $: since $ AB \parallel CD $, alternate interior angles? Wait, maybe $ \angle x = 65^\circ $? Wait, no, let's re-examine.

Wait, the problem is to find $ a $, $ b $, $ \angle x $. So:

- $ a $: length of $ AD $. In parallelogram $ ABCD $, $ AD = BC $ (opposite sides of parallelogram are equal). $ BC = 6 $ cm, so $ a = 6 $ cm.
- $ b $: length of $ BD $? No, $ b $ is a side? Wait, no, $ b $ is a segment, maybe $ AB = 5 $ cm, so $ b = 5 $ cm? Wait, the diagram shows $ AB $ side with length 5 cm? Wait, the triangle has 5 cm and 6 cm. So $ a = 6 $ cm (AD = BC), $ b = 5 $ cm (AB = CD), and $ \angle x $: in triangle $ BCD $, angle at $ C $ is $ \angle x $, and since $ AB \parallel CD $, angle $ \angle ABC = \angle ADC $, but maybe $ \angle x = 65^\circ $? Wait, maybe I made a mistake. Let's correct:

In parallelogram $ ABCD $:

- $ AD = BC = 6 $ cm (opposite sides equal) ⇒ $ a = 6 $ cm.
- $ AB = CD = 5 $ cm (opposite sides equal) ⇒ $ b = 5 $ cm.
- For $ \angle x $: in triangle $ BCD $, or using the angle at $ B $ is $ 65^\circ $, so since $ AB \parallel CD $, $ \angle x = 65^\circ $ (alternate interior angles).

So:

$ a = 6 $ cm, $ b = 5 $ cm, $ \angle x = 65^\circ $.

## Step 2: Verify
Check parallelogram properties: opposite sides equal, so $ AD = BC = 6 $, $ AB = CD = 5 $. Angles: alternate interior angles equal, so $ \angle x = 65^\circ $.

# Answer:
$ a = 6 $ cm, $ b = 5 $ cm, $ \angle x = 65^\circ $

### Problem 1 (2)
# Explanation:
## Step 1: Recall parallelogram properties
In parallelogram $ ABCD $, opposite sides are equal, so $ AB = CD = 6 $ cm (so $ c = AB = 6 $ cm). Adjacent angles are supplementary, so $ \angle A = 120^\circ $, so $ \angle z = 180 - 120 = 60^\circ $? Wait, no: $ \angle A = 120^\circ $, so $ \angle C = \angle A = 120^\circ $? No, wait adjacent angles: $ \angle A + \angle B = 180^\circ $, but $ \angle z $ is at $ C $, which is adjacent to $ \angle B $. Wait, $ AD \parallel BC $, so $ \angle A + \angle B = 180^\circ $, but $ \angle z $ is $ \angle C $, which is equal to $ \angle A $? No, wait in parallelogram, opposite angles are equal, so $ \angle A = \angle C $, $ \angle B = \angle D $. Wait, $ \angle A = 120^\circ $, so $ \angle C = \angle z = 120^\circ $? No, that can't be. Wait, no: $ AD = 5 $ cm, $ AB = 6 $ cm. So $ c = AB = 6 $ cm (since $ CD = AB $ in parallelogram). Then $ \angle y $: $ \angle A = 120^\circ $, so $ \angle D = \angle y $, and adjacent angles are supplementary, so $ \angle y = 180 - 120 = 60^\circ $? Wait, no: $ \angle A $ and $ \angle D $ are adjacent, so $ \angle A + \angle D = 180^\circ $, so $ \angle y = 180 - 120 = 60^\circ $. Then $ \angle z = \angle A = 120^\circ $? Wait, no, opposite angles: $ \angle A = \angle C = 120^\circ $, so $ \angle z = 120^\circ $. Wait, let's clarify:

- $ c $: in parallelogram $ ABCD $, $ AB = CD = 6 $ cm (since $ AB = 6 $ cm, so $ CD = c = 6 $ cm).
- $ \angle y $: $ \angle A = 120^\circ $, $ AD \parallel BC $, so $ \angle A + \angle D = 180^\circ $ (adjacent angles), so $ \angle y = \angle D = 180 - 120 = 60^\circ $.
- $ \angle z $: $ \angle C = \angle A = 120^\circ $ (opposite angles in parallelogram), so $ \angle z = 120^\circ $.

## Step 2: Verify
- $ c = AB = 6 $ cm (opposite sides equal).
- $ \angle y = 60^\circ $ (adjacent angles supplementary to $ \angle A $).
- $ \angle z = 120^\circ $ (opposite angles equal to $ \angle A $).

# Answer:
$ c = 6 $ cm, $ \angle y = 60^\circ $, $ \angle z = 120^\circ $

### Problem 2
#### (1)
# Explanation:
In a parallelogram, if $ \angle A = \angle C $, then the other pair of opposite angles must also be equal, i.e., $ \angle B = \angle D $. Because in a quadrilateral, sum of interior angles is $ 360^\circ $, so if $ \angle A = \angle C $, then $ \angle B = \angle D = \frac{360 - 2\angle A}{2} = 180 - \angle A $, which is the condition for a parallelogram (opposite angles equal). So the missing part is $ \angle B = \angle D $.

# Answer:
$ \angle B = \angle D $

#### (2)
# Explanation:
If the intersection point $ O $ of diagonals $ AC $ and $ BD $ has $ AO = CO $, then for it to be a parallelogram, the other diagonal must also bisect each other, i.e., $ BO = DO $. Because in a parallelogram, diagonals bisect each other. So the missing part is $ BO = DO $.

# Answer:
$ BO = DO $

#### (3)
# Explanation:
In a parallelogram, if one pair of opposite sides is equal ($ AB = CD $), then the other pair of opposite sides must also be equal, i.e., $ AD = BC $, or alternatively, the sides $ AB $ and $ CD $ are parallel ($ AB \parallel CD $). But the standard condition is that if one pair of opposite sides is equal and parallel, or both pairs of opposite sides are equal. So the missing part is $ AD = BC $ (or $ AB \parallel CD $, but more accurately, both pairs of opposite sides equal, so $ AD = BC $).

# Answer:
$ AD = BC $ (or $ AB \parallel CD $, but typically $ AD = BC $)

#### (4)
# Explanation:
In a parallelogram, if one pair of opposite sides is parallel ($ AD \parallel BC $), then the other pair of opposite sides must also be parallel ($ AB \parallel CD $), or the other pair of opposite sides is equal ($ AB = CD $). But the standard condition is that both pairs of opposite sides are parallel, or one pair is parallel and equal. So the missing part is $ AB \parallel CD $ (or $ AB = CD $).

# Answer:
$ AB \parallel CD $ (or $ AB = CD $)

### Problem 3
#### (1)
# Explanation:
A parallelogram with equal diagonals is a rectangle. Because in a parallelogram, if diagonals are equal, then by the converse of the rectangle theorem, it is a rectangle (all angles are right angles, and diagonals are equal in a rectangle).

# Answer:
Rectangle

#### (2)
# Explanation:
A parallelogram with adjacent sides equal is a rhombus. Because in a parallelogram, if adjacent sides are equal, then all sides are equal (since opposite sides are equal in a parallelogram), which is the definition of a rhombus.

# Answer:
Rhombus

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QUESTION IMAGE

Question

Explanation:

Problem 1 (1)

Step 1: Recall properties of parallelograms

Step 2: Verify

Step 1: Recall parallelogram properties

Step 2: Verify

Answer:

Problem 1 (2)