298336
If all the angles of a triangle are acute, the triangle is called:
1 Obtuse-angled
2 Acute-angled
3 Right-angled
4 None of these
Explanation:
Acute-angled An equilateral triangle is a triangle with all three angles and sides equal. If all a triangle's angles are equal, they all must measure 60 degrees, adding up to 180 degrees total. Since all angles are acute, or less than 90 degrees, an equilateral triangle is also an acute triangle.
THE TRIANGLE AND ITS PROPERTIES
298337
In a \(\triangle \text{ABC},\) if \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C},\) then th s measure of the smallest angle is:
1 90°
2 60°
3 40°
4 30°
Explanation:
30° We have, \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C}\) \(\therefore 3\angle \text{B}=2\angle \text{A}\) and \(6\angle \text{C}=2\angle \text{A}\) \(\Rightarrow \angle \text{B}=\frac{2}{3}\angle \text{A}\) and \(\angle \text{C}=\frac{2}{6}\angle \text{A}=\frac{1}{3}\angle \text{A}\) Now, \(\angle \text{A}+\angle \text{B}+\angle \text{C}=180^\circ\) [Angle sum property of triangle] \(\Rightarrow \angle \text{A}+\frac{2}{3}\angle \text{A}+\frac{1}{3}\angle \text{A}=180^\circ\) \(\Rightarrow 3\angle \text{A}+2\angle \text{A}+\angle \text{A}=180^\circ\times3\) \(\Rightarrow 6\angle \text{A}=540^\circ\) \(\Rightarrow \angle \text{A}=90^\circ\) \(\therefore\) Smallest angle \(=\angle \text{C}=\frac{1}{3}\angle \text{A}=\frac{1}{3}\times90^\circ\) \(=30^\circ\) Hence, the correct answer is option (d).
THE TRIANGLE AND ITS PROPERTIES
298338
In Figure. PQ = PS. The value of x is:7
1 35°
2 45°
3 55°
4 70°
Explanation:
45° In \(\triangle\text{PQS},\) \(110^{\circ}+\angle\text{1}=180^{\circ}\) [linear pair of angles] \(\Rightarrow \ ?\text{1}=180^{\circ}-110^{\circ}\) \(\Rightarrow\angle\text{1}=70^{\circ}\) 8 Also, \( \ ?\text{1}=\angle\text{2}=70^{\circ}\) \([\because\text{PQ}=\text{PS}]\) As we know, the measures of any eterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. \(\therefore \ \angle2=\text{x}+25^{\circ}\) \(\Rightarrow 70^{\circ}=\text{x}+25^{\circ}\) \(\because\angle2=70^{0}\) \(\Rightarrow\text{x}=70^{\circ}-25^{\circ}\) \(\Rightarrow \ \text{x}=45^{\circ}\)
THE TRIANGLE AND ITS PROPERTIES
298341
If for \(\triangle \text{ABC}\) and \(\triangle\text{DEF},\) the correspondence CAB ↔ EDF gives a congruence, then which of the following is not true?
1 AC = DE
2 AB = EF
3 \(\angle\text{A}=\angle\text{D}\)
4 \(\angle\text{C}=\angle\text{E}\)
Explanation:
AB = EF Two figures are said to be congruent, if the trace copy of figure 1 fits exactly on that of: 9 Now, if \(\triangle\text{ABC}\) and \(\triangle\text{DEF}\) are congruent, then AB = DF, BC = EF AC = DE, \(\angle\text{A}=\angle\text{D}\) \(\angle\text{B}=\angle\text{F},\) \(\angle\text{ C}=\angle\text{E}\) Hence, option (b) is not true.
298336
If all the angles of a triangle are acute, the triangle is called:
1 Obtuse-angled
2 Acute-angled
3 Right-angled
4 None of these
Explanation:
Acute-angled An equilateral triangle is a triangle with all three angles and sides equal. If all a triangle's angles are equal, they all must measure 60 degrees, adding up to 180 degrees total. Since all angles are acute, or less than 90 degrees, an equilateral triangle is also an acute triangle.
THE TRIANGLE AND ITS PROPERTIES
298337
In a \(\triangle \text{ABC},\) if \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C},\) then th s measure of the smallest angle is:
1 90°
2 60°
3 40°
4 30°
Explanation:
30° We have, \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C}\) \(\therefore 3\angle \text{B}=2\angle \text{A}\) and \(6\angle \text{C}=2\angle \text{A}\) \(\Rightarrow \angle \text{B}=\frac{2}{3}\angle \text{A}\) and \(\angle \text{C}=\frac{2}{6}\angle \text{A}=\frac{1}{3}\angle \text{A}\) Now, \(\angle \text{A}+\angle \text{B}+\angle \text{C}=180^\circ\) [Angle sum property of triangle] \(\Rightarrow \angle \text{A}+\frac{2}{3}\angle \text{A}+\frac{1}{3}\angle \text{A}=180^\circ\) \(\Rightarrow 3\angle \text{A}+2\angle \text{A}+\angle \text{A}=180^\circ\times3\) \(\Rightarrow 6\angle \text{A}=540^\circ\) \(\Rightarrow \angle \text{A}=90^\circ\) \(\therefore\) Smallest angle \(=\angle \text{C}=\frac{1}{3}\angle \text{A}=\frac{1}{3}\times90^\circ\) \(=30^\circ\) Hence, the correct answer is option (d).
THE TRIANGLE AND ITS PROPERTIES
298338
In Figure. PQ = PS. The value of x is:7
1 35°
2 45°
3 55°
4 70°
Explanation:
45° In \(\triangle\text{PQS},\) \(110^{\circ}+\angle\text{1}=180^{\circ}\) [linear pair of angles] \(\Rightarrow \ ?\text{1}=180^{\circ}-110^{\circ}\) \(\Rightarrow\angle\text{1}=70^{\circ}\) 8 Also, \( \ ?\text{1}=\angle\text{2}=70^{\circ}\) \([\because\text{PQ}=\text{PS}]\) As we know, the measures of any eterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. \(\therefore \ \angle2=\text{x}+25^{\circ}\) \(\Rightarrow 70^{\circ}=\text{x}+25^{\circ}\) \(\because\angle2=70^{0}\) \(\Rightarrow\text{x}=70^{\circ}-25^{\circ}\) \(\Rightarrow \ \text{x}=45^{\circ}\)
THE TRIANGLE AND ITS PROPERTIES
298341
If for \(\triangle \text{ABC}\) and \(\triangle\text{DEF},\) the correspondence CAB ↔ EDF gives a congruence, then which of the following is not true?
1 AC = DE
2 AB = EF
3 \(\angle\text{A}=\angle\text{D}\)
4 \(\angle\text{C}=\angle\text{E}\)
Explanation:
AB = EF Two figures are said to be congruent, if the trace copy of figure 1 fits exactly on that of: 9 Now, if \(\triangle\text{ABC}\) and \(\triangle\text{DEF}\) are congruent, then AB = DF, BC = EF AC = DE, \(\angle\text{A}=\angle\text{D}\) \(\angle\text{B}=\angle\text{F},\) \(\angle\text{ C}=\angle\text{E}\) Hence, option (b) is not true.
298336
If all the angles of a triangle are acute, the triangle is called:
1 Obtuse-angled
2 Acute-angled
3 Right-angled
4 None of these
Explanation:
Acute-angled An equilateral triangle is a triangle with all three angles and sides equal. If all a triangle's angles are equal, they all must measure 60 degrees, adding up to 180 degrees total. Since all angles are acute, or less than 90 degrees, an equilateral triangle is also an acute triangle.
THE TRIANGLE AND ITS PROPERTIES
298337
In a \(\triangle \text{ABC},\) if \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C},\) then th s measure of the smallest angle is:
1 90°
2 60°
3 40°
4 30°
Explanation:
30° We have, \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C}\) \(\therefore 3\angle \text{B}=2\angle \text{A}\) and \(6\angle \text{C}=2\angle \text{A}\) \(\Rightarrow \angle \text{B}=\frac{2}{3}\angle \text{A}\) and \(\angle \text{C}=\frac{2}{6}\angle \text{A}=\frac{1}{3}\angle \text{A}\) Now, \(\angle \text{A}+\angle \text{B}+\angle \text{C}=180^\circ\) [Angle sum property of triangle] \(\Rightarrow \angle \text{A}+\frac{2}{3}\angle \text{A}+\frac{1}{3}\angle \text{A}=180^\circ\) \(\Rightarrow 3\angle \text{A}+2\angle \text{A}+\angle \text{A}=180^\circ\times3\) \(\Rightarrow 6\angle \text{A}=540^\circ\) \(\Rightarrow \angle \text{A}=90^\circ\) \(\therefore\) Smallest angle \(=\angle \text{C}=\frac{1}{3}\angle \text{A}=\frac{1}{3}\times90^\circ\) \(=30^\circ\) Hence, the correct answer is option (d).
THE TRIANGLE AND ITS PROPERTIES
298338
In Figure. PQ = PS. The value of x is:7
1 35°
2 45°
3 55°
4 70°
Explanation:
45° In \(\triangle\text{PQS},\) \(110^{\circ}+\angle\text{1}=180^{\circ}\) [linear pair of angles] \(\Rightarrow \ ?\text{1}=180^{\circ}-110^{\circ}\) \(\Rightarrow\angle\text{1}=70^{\circ}\) 8 Also, \( \ ?\text{1}=\angle\text{2}=70^{\circ}\) \([\because\text{PQ}=\text{PS}]\) As we know, the measures of any eterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. \(\therefore \ \angle2=\text{x}+25^{\circ}\) \(\Rightarrow 70^{\circ}=\text{x}+25^{\circ}\) \(\because\angle2=70^{0}\) \(\Rightarrow\text{x}=70^{\circ}-25^{\circ}\) \(\Rightarrow \ \text{x}=45^{\circ}\)
THE TRIANGLE AND ITS PROPERTIES
298341
If for \(\triangle \text{ABC}\) and \(\triangle\text{DEF},\) the correspondence CAB ↔ EDF gives a congruence, then which of the following is not true?
1 AC = DE
2 AB = EF
3 \(\angle\text{A}=\angle\text{D}\)
4 \(\angle\text{C}=\angle\text{E}\)
Explanation:
AB = EF Two figures are said to be congruent, if the trace copy of figure 1 fits exactly on that of: 9 Now, if \(\triangle\text{ABC}\) and \(\triangle\text{DEF}\) are congruent, then AB = DF, BC = EF AC = DE, \(\angle\text{A}=\angle\text{D}\) \(\angle\text{B}=\angle\text{F},\) \(\angle\text{ C}=\angle\text{E}\) Hence, option (b) is not true.
298336
If all the angles of a triangle are acute, the triangle is called:
1 Obtuse-angled
2 Acute-angled
3 Right-angled
4 None of these
Explanation:
Acute-angled An equilateral triangle is a triangle with all three angles and sides equal. If all a triangle's angles are equal, they all must measure 60 degrees, adding up to 180 degrees total. Since all angles are acute, or less than 90 degrees, an equilateral triangle is also an acute triangle.
THE TRIANGLE AND ITS PROPERTIES
298337
In a \(\triangle \text{ABC},\) if \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C},\) then th s measure of the smallest angle is:
1 90°
2 60°
3 40°
4 30°
Explanation:
30° We have, \(2\angle \text{A}=3\angle \text{B}=6\angle \text{C}\) \(\therefore 3\angle \text{B}=2\angle \text{A}\) and \(6\angle \text{C}=2\angle \text{A}\) \(\Rightarrow \angle \text{B}=\frac{2}{3}\angle \text{A}\) and \(\angle \text{C}=\frac{2}{6}\angle \text{A}=\frac{1}{3}\angle \text{A}\) Now, \(\angle \text{A}+\angle \text{B}+\angle \text{C}=180^\circ\) [Angle sum property of triangle] \(\Rightarrow \angle \text{A}+\frac{2}{3}\angle \text{A}+\frac{1}{3}\angle \text{A}=180^\circ\) \(\Rightarrow 3\angle \text{A}+2\angle \text{A}+\angle \text{A}=180^\circ\times3\) \(\Rightarrow 6\angle \text{A}=540^\circ\) \(\Rightarrow \angle \text{A}=90^\circ\) \(\therefore\) Smallest angle \(=\angle \text{C}=\frac{1}{3}\angle \text{A}=\frac{1}{3}\times90^\circ\) \(=30^\circ\) Hence, the correct answer is option (d).
THE TRIANGLE AND ITS PROPERTIES
298338
In Figure. PQ = PS. The value of x is:7
1 35°
2 45°
3 55°
4 70°
Explanation:
45° In \(\triangle\text{PQS},\) \(110^{\circ}+\angle\text{1}=180^{\circ}\) [linear pair of angles] \(\Rightarrow \ ?\text{1}=180^{\circ}-110^{\circ}\) \(\Rightarrow\angle\text{1}=70^{\circ}\) 8 Also, \( \ ?\text{1}=\angle\text{2}=70^{\circ}\) \([\because\text{PQ}=\text{PS}]\) As we know, the measures of any eterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. \(\therefore \ \angle2=\text{x}+25^{\circ}\) \(\Rightarrow 70^{\circ}=\text{x}+25^{\circ}\) \(\because\angle2=70^{0}\) \(\Rightarrow\text{x}=70^{\circ}-25^{\circ}\) \(\Rightarrow \ \text{x}=45^{\circ}\)
THE TRIANGLE AND ITS PROPERTIES
298341
If for \(\triangle \text{ABC}\) and \(\triangle\text{DEF},\) the correspondence CAB ↔ EDF gives a congruence, then which of the following is not true?
1 AC = DE
2 AB = EF
3 \(\angle\text{A}=\angle\text{D}\)
4 \(\angle\text{C}=\angle\text{E}\)
Explanation:
AB = EF Two figures are said to be congruent, if the trace copy of figure 1 fits exactly on that of: 9 Now, if \(\triangle\text{ABC}\) and \(\triangle\text{DEF}\) are congruent, then AB = DF, BC = EF AC = DE, \(\angle\text{A}=\angle\text{D}\) \(\angle\text{B}=\angle\text{F},\) \(\angle\text{ C}=\angle\text{E}\) Hence, option (b) is not true.
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