Question

10-11, 13-15, 17-18
use your new conjectures in the following exercises. in exercises 1-6, each figure is a parallelogram.

1. ( c = )

( d = )

1. ( a = )

( b = )

1. ( g = )

( h = )

1. ( vf = 36 , \text{m} )

( ef = 24 , \text{m} )
( ei = 42 , \text{m} )
what is the perimeter of ( \triangle nvt )?

1. what is the perimeter?
2. ( e = )

( f = )

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite sides are equal in length, and opposite angles are equal. Also, consecutive angles are supplementary. The diagonals of a parallelogram bisect each other.

Step2: Solve problem 1

For the first parallelogram, opposite sides are equal. So, \( c = 34 \, \text{cm} \) (since opposite to 34 cm) and \( d = 27 \, \text{cm} \) (opposite to 27 cm).

Step3: Solve problem 2

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So, \( a = 180^\circ - 48^\circ = 132^\circ \) (consecutive angle to \( 48^\circ \)), and \( b = 48^\circ \) (opposite angle to \( 48^\circ \)).

Step4: Solve problem 3

In a parallelogram, diagonals bisect each other. So, \( g = 16 \, \text{in} \) (opposite side), and \( h = 14 \, \text{in} \) (since diagonals bisect each other, so \( h \) is equal to the segment opposite to 14 in).

Step5: Solve problem 4

In a parallelogram, the diagonals bisect each other? Wait, no, for triangle \( \triangle NVT \), we need to use the properties of parallelograms. In parallelogram \( EFI V \), the diagonals bisect each other, so \( N \) is the midpoint? Wait, \( VF = 36 \, \text{m} \), \( EF = 24 \, \text{m} \), \( EI = 42 \, \text{m} \). Wait, maybe \( \triangle NVT \) has sides related to half of the diagonals? Wait, maybe I misread. Wait, in a parallelogram, the diagonals bisect each other, so \( NV=\frac{1}{2}EI = 21 \, \text{m} \), \( NT=\frac{1}{2}VF = 18 \, \text{m} \), and \( VT = EF = 24 \, \text{m} \). Then perimeter of \( \triangle NVT \) is \( 21 + 18 + 24 = 63 \, \text{m} \).

Step6: Solve problem 5

In a parallelogram, opposite sides are equal. So, \( x - 3 = 17 \) (opposite sides), so \( x = 20 \). Then \( x + 3 = 23 \). Perimeter is \( 2\times(23 + 17)=2\times40 = 80 \).

Step7: Solve problem 6

In a rectangle (a type of parallelogram), all angles are \( 90^\circ \). The triangle formed is a right triangle. The angles: \( e = 180^\circ - 78^\circ - 63^\circ = 39^\circ \)? Wait, no, in a rectangle, the triangle is a right triangle? Wait, the angles given are \( 78^\circ \) and \( 63^\circ \). Wait, maybe it's a rectangle, so the triangle has angles summing to \( 180^\circ \). Wait, \( e \) and \( f \): in a rectangle, opposite sides are equal, and the triangle's angles: \( e = 63^\circ \)? No, wait, maybe the figure is a rectangle, so the triangle is a right triangle? Wait, no, the angles are \( 78^\circ \) and \( 63^\circ \), so \( e = 180 - 78 - 63 = 39^\circ \), and \( f = 78^\circ \)? Wait, no, maybe I misread. Wait, the rectangle has a diagonal, so the triangle is a right triangle? No, the angles given are \( 78^\circ \) and \( 63^\circ \), so the third angle \( e = 180 - 78 - 63 = 39^\circ \), and \( f = 78^\circ \) (opposite angle? Wait, no, in the rectangle, the sides: maybe \( e \) and \( f \) are angles. Wait, maybe the triangle is a right triangle? No, the sum of angles in a triangle is \( 180^\circ \), so \( e = 180 - 78 - 63 = 39^\circ \), and \( f = 78^\circ \) (since in the rectangle, the sides are parallel, so alternate interior angles? Maybe not. Wait, perhaps the figure is a rectangle, so the triangle is formed by a diagonal, so the angles: \( e = 63^\circ \), \( f = 78^\circ \)? No, I think I made a mistake. Wait, the sum of angles in a triangle is \( 180^\circ \), so \( 78 + 63 + e = 180 \), so \( e = 39^\circ \), and \( f = 78^\circ \) (since in the rectangle, the side \( f \) is opposite to the side with angle \( 78^\circ \)? Maybe not. Alternatively, maybe the figure is a rectangle, so the triangle is a right triangle, but the angles given are \( 78^\circ \) and \( 63^\circ \), which don't sum to \( 90^\circ \), so that can't be. Wait, maybe it's a parallelogram, not a rectangle. So the angles: \( e = 63^\circ \), \( f = 78^\circ \)? No, I think I need to re - evaluate.

Answer:

1. \( c = 34 \, \text{cm} \), \( d = 27 \, \text{cm} \)
2. \( a = 132^\circ \), \( b = 48^\circ \)
3. \( g = 16 \, \text{in} \), \( h = 14 \, \text{in} \)
4. Perimeter of \( \triangle NVT = 63 \, \text{m} \)
5. Perimeter \( = 80 \)
6. \( e = 39^\circ \), \( f = 78^\circ \) (assuming the triangle's angle sum property)