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Chapter 5: Problem 145

Given any \(\triangle A B C,\) if we extend the side \(\underline{B C}\) beyond \(C\) to a point \(X,\) then the "exterior angle" \(\angle A C X\) at \(C\) is greater than each of the "two interior opposite angles" \(\angle A\) and \(\angle B\).

Short Answer

Expert verified
The exterior angle \(\angle ACX\) is greater than each of \(\angle A\) and \(\angle B\).

Step by step solution

01

Understanding the Angle Properties

In any triangle, an exterior angle is formed by extending one of the sides. According to the exterior angle theorem, the exterior angle is equal to the sum of the two opposite interior angles.
02

Identifying the Angles Involved

In \(\triangle ABC\), extending the side \(BC\) to a new point \(X\) creates the exterior angle \(\angle ACX\). The two opposite interior angles are \(\angle A\) and \(\angle B\). According to the theorem, \(\angle ACX = \angle A + \angle B\).
03

Applying the Exterior Angle Theorem

The theorem states that an exterior angle (\(\angle ACX\)) of a triangle is greater than either of the non-adjacent interior angles (\(\angle A\) and \(\angle B\)). Therefore, \(\angle ACX > \angle A\) and \(\angle ACX > \angle B\).
04

Conclusion from the Theorem

Since \(\angle ACX\) is the sum of \(\angle A\) and \(\angle B\), it must be greater than each angle individually. Thus, the exterior angle \(\angle ACX\) is greater than \(\angle A\), and it is also greater than \(\angle B\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle Geometry
Triangles are one of the most fundamental shapes in geometry, and they are everywhere in our world. A triangle is a simple polygon with three edges and three vertices. The three sides of a triangle can be of different lengths, and the measures of the angles can vary. However, certain properties always hold true for triangles, such as the sum of the interior angles always being 180 degrees.

Another important aspect of triangle geometry is understanding how angles are formed and interact. Triangles have both interior and exterior angles, which play a key role in many geometric theorems and properties. These angles are essential for solving various geometric problems and proofs. By focusing on the key properties of triangles, you can gain a deeper understanding of geometric principles.
Angle Properties
Angles are fundamental in understanding geometrical shapes and their properties. In the context of triangles, angles are crucial because they define the shape and size of the triangle. Let's summarize some of the basic angle properties relevant in triangles:
  • The sum of the interior angles in any triangle is always 180 degrees.
  • Exterior angles are formed by extending one side of the triangle.
  • According to the Exterior Angle Theorem, an exterior angle is equal to the sum of the two opposite interior angles.
  • Because an exterior angle is the sum of the non-adjacent interior angles, it is always greater than either of those individual angles.
Understanding these properties helps in solving many geometric problems and is essential for grasping more complex geometric theorems.
Interior and Exterior Angles
Interior and exterior angles are key concepts in triangle geometry. An interior angle is formed within the triangle by two adjacent sides. Each triangle has three interior angles. On the other hand, an exterior angle is formed by one side of the triangle and the extension of an adjacent side.

The Exterior Angle Theorem is especially significant. It states that the measure of an exterior angle is equal to the sum of its two non-adjacent interior angles. This theorem helps in comprehending how the angle measures relate to each other, strengthening your problem-solving skills.

For example, consider \\(\triangle ABC\)\, where we extend \(BC\)\, forming an exterior angle \(\angle ACX\). According to the theorem, \(\angle ACX = \angle A + \angle B\). This relation shows why \(\angle ACX\) is always greater than \(\angle A\) or \(\angle B\), as it's formed by adding them together. Recognizing these relations will make it easier to solve geometric problems and prove various theorems.

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Most popular questions from this chapter

Find the exact length (in terms of \(\pi)\) (i) of a semicircle of radius \(r\); (ii) of a quarter circle of radius \(r\); (iii) of the length of an arc of a circle of radius \(r\) that subtends an angle \(\theta\) radians at the centre. \(\wedge\) In the next problem we follow a similar sequence of steps to conclude that the quotient \(L=\frac{\text { area of circle of radius } r}{r^{2}}\) is also constant. The surprise lies in the fact that this different constant is so closely related to the previous constant \(K\).

Suppose that in \(\triangle A B C, \angle C=\angle A+\angle B\). Prove that \(C\) lies on the circle with diameter \(\underline{A B}\). (In particular, if the angles of \(\triangle A B C\) add to a straight angle, and \(\angle A C B\) is a right angle, then \(C\) lies on the circle with diameter \(\underline{A B}\).)

Consider the cube with edges of length 2 running parallel to the coordinate axes, with its centre at the origin \((0,0,0),\) and with opposite corners at (1,1,1) and (-1,-1,-1) . The \(x-, y-,\) and \(z\) -axes, and the \(x y\) -, \(y z-,\) and \(z x\) -planes cut this cube into eight unit cubes \(-\) one sitting in each octant. (i) Let \(A=(0,0,1), B=(1,0,0), C=(0,1,0), W=(1,1,1) .\) Describe the solid \(A B C W\) (ii) Let \(D=(-1,0,0), X=(-1,1,1) .\) Describe the solid \(A C D X\). (iii) Let \(E=(0,-1,0), Y=(-1,-1,1) .\) Describe the solid \(A D E Y\). (iv) Let \(Z=(1,-1,1) .\) Describe the solid \(A E B Z\). (v) Let \(F=(0,0,-1)\) and repeat steps (i)-(iv) to obtain the four mirror image solids which lie beneath the \(x y\) -plane. (vi) Describe the solid \(A B C D E F\) which is surrounded by the eight identical solids in (i)-(v).

(a) Find the exact area (in terms of \(\pi\) ) (i) of a semicircle of radius \(r\); (ii) of a quarter circle of radius \(r\) (iii) of a sector of a circle of radius \(r\) that subtends an angle \(\theta\) radians at the centre. (b) Find the area of a sector of a circle of radius \(1,\) whose total perimeter (including the two radii) is exactly half that of the circle itself.

(a) Describe the cross-sections obtained by cutting such a double cone by a horizontal plane (i.e. a plane perpendicular to the \(z\) -axis). What if the cutting plane is the \(x y\) -plane? (b)(i) Describe the cross-section obtained by cutting such a cone by a vertical plane through the origin, or apex. (ii) What cross-section is obtained if the cutting plane passes through the apex, but is not vertical? (c) Give a qualitative description of the curve obtained as a cross-section of the cone if we cut the cone by a plane which is parallel to a generator: e.g. the plane \(y-r z=c\). (i) What happens if \(c=0 ?\) (ii) Now assume the cutting plane is parallel to a generator, but does not pass through the apex of the cone - so we may assume that the plane cuts only the bottom half of the cone. Insert a small sphere inside the bottom half of the cone and above the cutting plane, and inflate the sphere as much as possible \(-\) until it touches the cone around a horizontal circle (the "contact circle with the cone"), and touches the plane at a single point \(F\). Let the horizontal plane of the "contact circle with the cone" meet the cutting plane in the line \(m\). Prove that each point of the cross-sectional curve is equidistant from the point \(F\) and from the line \(m-\) and so is a parabola. (d)(i) Give a qualitative description of the curve obtained as a cross-section of the cone if we cut the cone by a plane which is less steep than a generator, but does not pass through the apex \(-\) and so cuts right across the cone. (ii) We may assume that the plane cuts only the bottom half of the cone. Insert a small sphere inside the bottom half of the cone and above the cutting plane (i.e. on the same side of the cutting plane as the apex of the cone), and inflate the sphere as much as possible \(-\) until it touches the cone around a horizontal circle, and touches the plane at a single point \(F\). Let the horizontal plane of the contact circle meet the cutting plane in the line \(m\). Prove that, for each point \(X\) on the cross-sectional curve, the ratio "distance from \(X\) to \(F ":\) "distance from \(X\) to \(m "=e: 1\) is constant, with \(e<1,\) and so is an ellipse. (e)(i) Give a qualitative description of the curve obtained as a cross-section if we cut the cone by a plane which is steeper than a generator, but does not pass through the apex (and hence cuts both halves of the cone)? (ii) We can be sure that the plane cuts the bottom half of the cone (as well as the top half). Insert a small sphere inside the bottom half of the cone and on the same side of the cutting plane as the apex, and inflate the sphere as much as possible \(-\) until it touches the cone around a horizontal circle, and touches the plane at a single point \(F\). Let the horizontal plane of the contact circle meet the cutting plane in the line \(m\). Prove that, for each point \(X\) on the cross-sectional curve, the ratio "distance from \(X\) to \(F ":\) "distance from \(X\) to \(m "=e: 1\) is constant, with \(e>1,\) and so is a hyperbola.
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