Math / Geometry

Question1. Which of the following measures of three angles can be those of a triangle? 1) $40^{\circ}, 60^{\circ}, 80^{\circ}$ 2) $90^{\circ}, 60^{\circ}, 20^{\circ}$ 3) $100^{\circ}, 80^{\circ}, 50^{\circ}$ 4) $40^{\circ}, 80^{\circ}, 90^{\circ}$
2. The measures of the four angles of a quadrilateral are in the ratio of 12.3:4. What is the measure of the greatest angle of quadrilateral? 1) $144^{\circ}$ 2) $135^{\circ}$ 3) $125^{\circ}$ 4) $150^{\circ}$
3. Difference between two complementary angles is $10^{\circ}$ . Calculate the vatues of both angles. 1) $45^{\circ}, 45^{\circ}$ 2) $40^{\circ}, 50^{\circ}$ 3) $50^{\circ}, 60^{\circ}$
4. What is the next row of numbers? 4) $100^{\circ}, 10^{\circ}$ \begin{tabular}{|l|l|l|} \hline 48 & 144 & 192 \\ \hline 38 & 114 & 152 \\ \hline 28 & 84 & 112 \\ \hline \end{tabular} 1) $18 \quad 54 \quad 72$ 2) $58 \quad 184 \quad 244$ 3) $68 \quad 204 \quad 272$
5. If $\angle 1=135^{\circ}, \angle 4=45^{\circ}$ . Then, find $\angle 2$ and $\angle 3$ , respectively 4) $68 \quad 214 \quad 292$ 1) $145^{\circ}, 35^{\circ}$ 2) $130^{\circ}, 50$ 3) $140^{\circ}, 40$ 4) $45^{\circ}, 135^{\circ}$
6. A quadrilateral $A B C D$ is inscribed in a circle. If $A B$ is parallel to $C D$ and $A C=B D$ , then the quadrilateral must be a 1) parallelogram 2) rhombus 3) trapezium - 4) None of these
7. Which of the following is true if $\mathrm{PQ}=\mathrm{RS}$ ? 1) $P Q+Q R=R S$ 2) $P R=Q S$ 3) $P Q+Q S=R S$ 4) $P Q-R S=Q R$

Studdy Solution

STEP 1

1. For a set of angles to form a triangle, their sum must be $180^{\circ}$ .
2. The sum of the measures of the angles in a quadrilateral is $360^{\circ}$ .
3. Complementary angles are two angles whose measures add up to $90^{\circ}$ .
4. To find patterns in sequences, we look for arithmetic or geometric progressions or other regular patterns.
5. For a quadrilateral inscribed in a circle with certain parallel sides and equal diagonals, specific properties of quadrilaterals can be used.
6. For equality of line segments, geometric properties and relationships are considered.

STEP 2

1. Verify if the given sets of angles can form a triangle.
2. Calculate the measures of the angles in the quadrilateral based on the given ratio.
3. Determine the values of two complementary angles given their difference.
4. Identify the pattern in the given number sequence and predict the next row.
5. Use given angles to find the remaining angles in a problem involving supplementary angles.
6. Determine the type of quadrilateral based on given properties.
7. Analyze the given geometric relationship to determine the true statement.

STEP 3

Verify if the given sets of angles can form a triangle by summing the angles in each option.
$\text{Option 1: } 40^{\circ} + 60^{\circ} + 80^{\circ} = 180^{\circ} \quad \text{(Valid)}$ $\text{Option 2: } 90^{\circ} + 60^{\circ} + 20^{\circ} = 170^{\circ} \quad \text{(Invalid)}$ $\text{Option 3: } 100^{\circ} + 80^{\circ} + 50^{\circ} = 230^{\circ} \quad \text{(Invalid)}$ $\text{Option 4: } 40^{\circ} + 80^{\circ} + 90^{\circ} = 210^{\circ} \quad \text{(Invalid)}$

STEP 4

Find the measure of the greatest angle in the quadrilateral given the ratio of 1:2:3:4.
Let the angles be $x, 2x, 3x, 4x$ . Therefore, $x + 2x + 3x + 4x = 360^{\circ}$

STEP 5

Solve for $x$ .
$10x = 360^{\circ}$ $x = 36^{\circ}$ Then the angles are $36^{\circ}, 72^{\circ}, 108^{\circ}, 144^{\circ}$ . The greatest angle is $144^{\circ}$ .

STEP 6

Determine the values of two complementary angles given the difference is $10^{\circ}$ .
Let the angles be $x$ and $90^{\circ} - x$ . Given, $x - (90^{\circ} - x) = 10^{\circ}$

STEP 7

Solve for $x$ .
$x - 90^{\circ} + x = 10^{\circ}$ $2x = 100^{\circ}$ $x = 50^{\circ}$ The angles are $50^{\circ}$ and $40^{\circ}$ .

STEP 8

Identify the pattern in the given number sequence and predict the next row.
Given rows: $48 \quad 144 \quad 192$ $38 \quad 114 \quad 152$ $28 \quad 84 \quad 112$

STEP 9

Identify the pattern by analyzing the differences between the numbers in each column.
Column differences: $48 - 38 = 10, \quad 38 - 28 = 10$ $144 - 114 = 30, \quad 114 - 84 = 30$ $192 - 152 = 40, \quad 152 - 112 = 40$
Next row (subtracting 10, 30, and 40 respectively): $28 - 10 = 18$ $84 - 30 = 54$ $112 - 40 = 72$ The next row is $18, 54, 72$ .

STEP 10

Use the given angles $\angle 1 = 135^{\circ}$ and $\angle 4 = 45^{\circ}$ to find $\angle 2$ and $\angle 3$ .
If the angles form a supplementary pair, we have, $\angle 1 + \angle 3 = 180^{\circ}$ $135^{\circ} + \angle 3 = 180^{\circ}$

STEP 11

Solve for $\angle 3$ .
$\angle 3 = 180^{\circ} - 135^{\circ} = 45^{\circ}$ $\text{Since } \angle 4 = 45^{\circ}, \angle 2 = 180^{\circ} - 45^{\circ} = 135^{\circ}$ The angles are $45^{\circ}$ and $135^{\circ}$ .

STEP 12

Determine the type of quadrilateral inscribed in a circle with given properties. $AB \parallel CD$ and $AC = BD$ imply:
When $AB \parallel CD$ and $AC = BD$ in a circle, the quadrilateral is a trapezium.

STEP 13

Analyze the geometric relationship given $\mathrm{PQ} = \mathrm{RS}$ .
Option 1: $PQ + QR = RS$ (Not always true) Option 2: $PR = QS$ (Depends on the quadrilateral properties) Option 3: $PQ + QS = RS$ (Not necessarily true) Option 4: $PQ - RS = QR$ (This could be true if the quadrilateral is cyclic)

STEP 14

Identify the correct relationship.
$PQ - RS = QR \quad \text{(Correct if $PQ = RS$)}$

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Find the measure of the greatest angle in the quadrilateral given the ratio of 1:2:3:4.
Let the angles be $x, 2x, 3x, 4x$ . Therefore, $x + 2x + 3x + 4x = 360^{\circ}$

STEP 5

Solve for $x$ .
$10x = 360^{\circ}$ $x = 36^{\circ}$ Then the angles are $36^{\circ}, 72^{\circ}, 108^{\circ}, 144^{\circ}$ . The greatest angle is $144^{\circ}$ .

STEP 6

Determine the values of two complementary angles given the difference is $10^{\circ}$ .
Let the angles be $x$ and $90^{\circ} - x$ . Given, $x - (90^{\circ} - x) = 10^{\circ}$

STEP 7

Solve for $x$ .
$x - 90^{\circ} + x = 10^{\circ}$ $2x = 100^{\circ}$ $x = 50^{\circ}$ The angles are $50^{\circ}$ and $40^{\circ}$ .

STEP 8

Identify the pattern in the given number sequence and predict the next row.
Given rows: $48 \quad 144 \quad 192$ $38 \quad 114 \quad 152$ $28 \quad 84 \quad 112$

STEP 9

STEP 10

Use the given angles $\angle 1 = 135^{\circ}$ and $\angle 4 = 45^{\circ}$ to find $\angle 2$ and $\angle 3$ .
If the angles form a supplementary pair, we have, $\angle 1 + \angle 3 = 180^{\circ}$ $135^{\circ} + \angle 3 = 180^{\circ}$

STEP 11

Solve for $\angle 3$ .
$\angle 3 = 180^{\circ} - 135^{\circ} = 45^{\circ}$ $\text{Since } \angle 4 = 45^{\circ}, \angle 2 = 180^{\circ} - 45^{\circ} = 135^{\circ}$ The angles are $45^{\circ}$ and $135^{\circ}$ .

STEP 12

Determine the type of quadrilateral inscribed in a circle with given properties. $AB \parallel CD$ and $AC = BD$ imply:
When $AB \parallel CD$ and $AC = BD$ in a circle, the quadrilateral is a trapezium.

STEP 13

Analyze the geometric relationship given $\mathrm{PQ} = \mathrm{RS}$ .
Option 1: $PQ + QR = RS$ (Not always true) Option 2: $PR = QS$ (Depends on the quadrilateral properties) Option 3: $PQ + QS = RS$ (Not necessarily true) Option 4: $PQ - RS = QR$ (This could be true if the quadrilateral is cyclic)

STEP 14

Identify the correct relationship.
$PQ - RS = QR \quad \text{(Correct if $PQ = RS$)}$

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Studdy Solution

STEP 1

STEP 2

STEP 3

STEP 4

Find the measure of the greatest angle in the quadrilateral given the ratio of 1:2:3:4.Let the angles be x,2x,3x,4xx, 2x, 3x, 4xx,2x,3x,4x. Therefore, x+2x+3x+4x=360∘ x + 2x + 3x + 4x = 360^{\circ} x+2x+3x+4x=360∘

STEP 5

Solve for xxx.10x=360∘ 10x = 360^{\circ} 10x=360∘ x=36∘ x = 36^{\circ} x=36∘ Then the angles are 36∘,72∘,108∘,144∘36^{\circ}, 72^{\circ}, 108^{\circ}, 144^{\circ}36∘,72∘,108∘,144∘. The greatest angle is 144∘144^{\circ}144∘.

STEP 6

Determine the values of two complementary angles given the difference is 10∘10^{\circ}10∘.Let the angles be xxx and 90∘−x90^{\circ} - x90∘−x. Given, x−(90∘−x)=10∘ x - (90^{\circ} - x) = 10^{\circ} x−(90∘−x)=10∘

STEP 7

Solve for xxx.x−90∘+x=10∘ x - 90^{\circ} + x = 10^{\circ} x−90∘+x=10∘ 2x=100∘ 2x = 100^{\circ} 2x=100∘ x=50∘ x = 50^{\circ} x=50∘ The angles are 50∘50^{\circ}50∘ and 40∘40^{\circ}40∘.

STEP 8

Identify the pattern in the given number sequence and predict the next row.Given rows: 48144192 48 \quad 144 \quad 192 48144192 38114152 38 \quad 114 \quad 152 38114152 2884112 28 \quad 84 \quad 112 2884112

STEP 9

STEP 10

STEP 11

STEP 12

Determine the type of quadrilateral inscribed in a circle with given properties. AB∥CDAB \parallel CDAB∥CD and AC=BDAC = BDAC=BD imply:When AB∥CDAB \parallel CDAB∥CD and AC=BDAC = BDAC=BD in a circle, the quadrilateral is a trapezium.

STEP 13

STEP 14

Identify the correct relationship.PQ−RS=QR(Correct if PQ=RS) PQ - RS = QR \quad \text{(Correct if $PQ = RS$)} PQ−RS=QR(Correct if PQ=RS)

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Find the measure of the greatest angle in the quadrilateral given the ratio of 1:2:3:4.
Let the angles be $x, 2x, 3x, 4x$ . Therefore, $x + 2x + 3x + 4x = 360^{\circ}$

Solve for $x$ .
$10x = 360^{\circ}$ $x = 36^{\circ}$ Then the angles are $36^{\circ}, 72^{\circ}, 108^{\circ}, 144^{\circ}$ . The greatest angle is $144^{\circ}$ .

Determine the values of two complementary angles given the difference is $10^{\circ}$ .
Let the angles be $x$ and $90^{\circ} - x$ . Given, $x - (90^{\circ} - x) = 10^{\circ}$

Solve for $x$ .
$x - 90^{\circ} + x = 10^{\circ}$ $2x = 100^{\circ}$ $x = 50^{\circ}$ The angles are $50^{\circ}$ and $40^{\circ}$ .

Identify the pattern in the given number sequence and predict the next row.
Given rows: $48 \quad 144 \quad 192$ $38 \quad 114 \quad 152$ $28 \quad 84 \quad 112$

Determine the type of quadrilateral inscribed in a circle with given properties. $AB \parallel CD$ and $AC = BD$ imply:
When $AB \parallel CD$ and $AC = BD$ in a circle, the quadrilateral is a trapezium.

Identify the correct relationship.
$PQ - RS = QR \quad \text{(Correct if $PQ = RS$)}$