Question

in δcde, m∠c = 45° and m∠d = 68°. which list has the sides of δcde in order from longest to shortest?
answer
○ ec, de, cd ○ de, ec, cd
○ de, cd, ec ○ cd, ec, de
○ ec, cd, de ○ cd, de, ec

Explanation:

Step1: Find the third angle

In a triangle, the sum of angles is \(180^\circ\). So, \(m\angle E = 180^\circ - 45^\circ - 68^\circ = 67^\circ\).

Step2: Relate angles to sides

In a triangle, the larger the angle, the longer the opposite side.

• Opposite \(\angle C = 45^\circ\) is \(DE\)
• Opposite \(\angle D = 68^\circ\) is \(EC\)
• Opposite \(\angle E = 67^\circ\) is \(CD\)

Order of angles from largest to smallest: \(\angle D (68^\circ) > \angle E (67^\circ) > \angle C (45^\circ)\)

Thus, order of sides (opposite these angles) from longest to shortest: \(EC\) (opposite \(\angle D\)), \(CD\) (opposite \(\angle E\)), \(DE\) (opposite \(\angle C\))? Wait, no—wait, let's correct. Wait, angle \(D\) is \(68^\circ\), opposite side is \(EC\); angle \(E\) is \(67^\circ\), opposite side is \(CD\); angle \(C\) is \(45^\circ\), opposite side is \(DE\). So angles: \(D (68^\circ) > E (67^\circ) > C (45^\circ)\). So sides opposite: \(EC\) (opposite \(D\)), \(CD\) (opposite \(E\)), \(DE\) (opposite \(C\)). Wait, but let's check the options. Wait, maybe I mixed up. Wait, triangle \(CDE\): vertices \(C\), \(D\), \(E\). So side opposite \(C\) is \(DE\), side opposite \(D\) is \(CE\) (or \(EC\)), side opposite \(E\) is \(CD\). So angle \(D = 68^\circ\) (largest), so side opposite (\(EC\)) is longest. Then angle \(E = 67^\circ\), side opposite (\(CD\)) is next. Then angle \(C = 45^\circ\), side opposite (\(DE\)) is shortest. Wait, but the options: let's list the options again.

Options:

1. EC, DE, CD

1. DE, EC, CD

1. DE, CD, EC

1. CD, EC, DE

1. EC, CD, DE

1. CD, DE, EC

Wait, let's recalculate angle \(E\): \(180 - 45 - 68 = 67^\circ\). So angles: \(D = 68^\circ\), \(E = 67^\circ\), \(C = 45^\circ\). So order of angles: \(D > E > C\). Therefore, sides opposite: \(EC\) (opposite \(D\)), \(CD\) (opposite \(E\)), \(DE\) (opposite \(C\)). So from longest to shortest: \(EC\), \(CD\), \(DE\). Wait, that's option 5: EC, CD, DE? Wait no, the fifth option is "EC, CD, DE"? Wait the user's options:

First column:

• EC, DE, CD

• DE, CD, EC

• EC, CD, DE

Second column:

• DE, EC, CD

• CD, EC, DE

• CD, DE, EC

Wait, maybe I made a mistake in side opposite. Let's label the triangle: vertices \(C\), \(D\), \(E\). So side \(CD\) is between \(C\) and \(D\), side \(DE\) between \(D\) and \(E\), side \(EC\) between \(E\) and \(C\). So angle at \(C\) is between \(CD\) and \(EC\), so side opposite angle \(C\) is \(DE\). Angle at \(D\) is between \(CD\) and \(DE\), so side opposite angle \(D\) is \(EC\). Angle at \(E\) is between \(DE\) and \(EC\), so side opposite angle \(E\) is \(CD\). So:

• Opposite \(\angle C\) (45°): \(DE\) (side \(DE\))

• Opposite \(\angle D\) (68°): \(EC\) (side \(EC\))

• Opposite \(\angle E\) (67°): \(CD\) (side \(CD\))

So since \(\angle D (68^\circ) > \angle E (67^\circ) > \angle C (45^\circ)\), then the sides opposite are \(EC > CD > DE\). So the order from longest to shortest is \(EC\), \(CD\), \(DE\). Which is the fifth option: EC, CD, DE.

Wait, but let's check again. Angle \(D\) is 68, so side opposite (EC) is longest. Angle \(E\) is 67, side opposite (CD) is next. Angle \(C\) is 45, side opposite (DE) is shortest. So EC (longest), CD (middle), DE (shortest). So the option is "EC, CD, DE" (fifth option).

Answer:

EC, CD, DE