sat suite question viewer

Geometry and Trigonometry Difficulty: Hard

In convex pentagon $A B C D E$ , segment $A B$ is parallel to segment $D E$ . The measure of angle $B$ is $139$ degrees, and the measure of angle $D$ is $174$ degrees. What is the measure, in degrees, of angle $C$ ?

Back question 185 of 268 Next

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268

Explanation

The correct answer is $47$ . It's given that the measure of angle $B$ is $139$ degrees. Therefore, the exterior angle formed by extending segment $A B$ at point $B$ has measure $180 minus 139$ , or $41$ , degrees. It's given that segment $A B$ is parallel to segment $D E$ . Extending segment $B C$ at point $C$ and extending segment $D E$ at point $D$ until the two segments intersect results in a transversal that intersects two parallel line segments. One of these intersection points is point $B$ , and let the other intersection point be point $X$ . Since segment $A B$ is parallel to segment $D E$ , alternate interior angles are congruent. Angle $C X D$ and the exterior angle formed by extending segment $A B$ at point $B$ are alternate interior angles. Therefore, the measure of angle $C X D$ is $41$ degrees. It's given that the measure of angle $D$ in pentagon $A B C D E$ is $174$ degrees. Therefore, angle $C D X$ has measure $180 minus 174$ , or $6$ , degrees. Since angle $C$ in pentagon $A B C D E$ is an exterior angle of triangle $C D X$ , it follows that the measure of angle $C$ is the sum of the measures of angles $C D X$ and $C X D$ . Therefore, the measure, in degrees, of angle $C$ is $6 plus 41$ , or $47$ .

Alternate approach: A line can be created that's perpendicular to segments $A B$ and $D E$ and passes through point $C$ . Extending segments $A B$ and $D E$ at points $B$ and $D$ , respectively, until they intersect this line yields two right triangles. Let these intersection points be point $X$ and point $Y$ , and the two right triangles be triangle $B X C$ and triangle $D Y C$ . It's given that the measure of angle $B$ is $139$ degrees. Therefore, angle $C B X$ has measure $180 minus 139$ , or $41$ , degrees. Since the measure of angle $C B X$ is $41$ degrees and the measure of angle $B X C$ is $90$ degrees, it follows that the measure of angle $X C B$ is $180 - 90 - 41$ , or $49$ , degrees. It's given that the measure of angle $D$ is $174$ degrees. Therefore, angle $Y D C$ has measure $180 minus 174$ , or $6$ , degrees. Since the measure of angle $Y D C$ is $6$ degrees and the measure of angle $C Y D$ is $90$ degrees, it follows that the measure of angle $D C Y$ is $180 - 90 - 6$ , or $84$ , degrees. Since angles $X C B$ , $D C Y$ , and angle $C$ in pentagon $A B C D E$ form segment $X Y$ , it follows that the sum of the measures of those angles is $180$ degrees. Therefore, the measure, in degrees, of angle $C$ is $180 - 49 - 84$ , or $47$ .